Arbitrary Lagrangian-Eulerian (ALE) Methods in Plasma Physics

Transcription

1 Czeh Tehnial University in Prague Faulty of Nulear Sienes and Physial Engineering Department of Physial Eletronis Arbitrary Lagrangian-Eulerian (ALE Methods in Plasma Physis Dotoral Thesis Author: Ing. Milan Kuhařík Supervisor: Do. Ing. Rihard Liska, CS. Adviser: Do. Ing. Jiří Limpouh, CS. Date: April 3, 6

2 Abstrat The omplete Arbitrary Lagrangian-Eulerian (ALE method, appliable to the fluid and laser plasma hydrodynamis, both in Cartesian and ylindrial geometries is presented. All parts of the ALE algorithm, i.e. Lagrangian solver, mesh smoothing, and onservative quantity remapping, in both geometries are fully desribed. The issues related to the laser plasma physis as plasma equation of state, thermal ondutivity, interation with laser beam, and sophistiated treatment of the boundary onditions neessary for the realisti laser plasma simulations are desribed. The omplete developed ALE ode is tested on a series of typial fluid problems to show its properties for the well known solutions. Finally, three sets of laser plasma simulations inspired by the real experiments are performed the interation of a laser beam with a massive target, ablative aeleration of small Aluminum dis flyer irradiated by a laser beam, and the high veloity impat of suh aelerated dis onto a massive Aluminum target. The standard Lagrangian simulation of the last, high veloity impat problem fails, and the omplete ALE methodology is required for this problem. Simulations of all types of problems show reasonable agreement with the experimental results.

3 Delaration I delare that this thesis is my own work, based on my personal study and researh. Information derived from the published and unpublished work of others has been aknowledged in the text, and a list of referenes is given. Milan Kuhařík Prague, April 3, 6

4 Contents Introdution 4. New Contributions of the Thesis Numerial Methods for Compressible Fluid Flows 7. Eulerian Methods Lagrangian Methods ALE Methods Hydrodynamial Codes for Plasma Simulations ALE Method in Cartesian Geometry 3 3. Lagrangian Solver Geometrial Struture Zonal Pressure Fore Subzonal Pressure Fore Visosity Fore Conservative Internal Energy Update Complete Algorithm of the Lagrangian Method Mesh Smoothing Tehniques Plain Averaging and Winslow Smoothing Advaned Rezoning Methods Conservative Remapping Algorithm Remapping of One Conservative Quantity Pieewise Linear Reonstrution Exat Numerial Integration Approximate Swept Numerial Integration Repair Remapping of All Quantities Numerial Testing of Remapping Methods Remapping in 3D Remapping with Changing Topology ALE Method in Cylindrial Geometry Lagrangian Solver Area-Weighted Differening Sheme Control Volume Method Mesh Smoothing Tehniques Conservative Remapping Algorithm Pieewise Linear Reonstrution Numerial Integration Exat Numerial Integration Swept

5 4.3.4 Repair Properties of Numerial ALE Method Conservativity Linearity Preservation, Order of Auray Numerial Tests Physial Aspets of ALE Simulations 6 6. Equation of State Heat Condutivity Interation with a Laser Beam Boundary Conditions Pressure Boundary Conditions Veloity Boundary Conditions Smoothing Mesh Boundary Conditions Boundary Conditions for Remapping Numerial Simulations of Laser Plasma Experiments 7 7. Derivation of Maximum Laser Intensity from Experimental Data Maximal Laser Intensity in D Maximal Laser Intensity in D Cartesian Geometry Maximal Laser Intensity in D Cylindrial Geometry Comparison of Maximal Laser Intensity Formulas Massive Target Irradiation by Laser Beam Ablative Flyer Aeleration by Laser Beam Purely Lagrangian Simulation in D D Simulation by Complete ALE Method Dis Flyer Impat Simulations Impat Simulations Started from D Uniform Initial Data Impat Simulations Started from D Interpolated Initial Data Energy Balane in the Simulations Conlusion 96 List of Seleted Important Publiations 98 Aknowledgment Bibliography 3

6 Chapter Introdution The first note about the numerial solving of partial differential equations appeared in 97, when the British sientist L. F. Rihardson introdued his first talk about weather foreast using this tehnique, done by hand, and published in 9 [8]. In general, this moment (although unsuessful due to the CFL problem is aknowledged as the beginning of the sientifi disipline, whih is nowadays alled the Computational Fluid Dynamis (CFD. From the time of Rihardson, not only the tehnial equipment allowing us to solve muh more extensive and omplex problems, but also theoretial knowledge essential for the omputations, have evolved. Today, we know lots of mathematial theorems desribing stability and auray of numerial methods, we know about the neessity of their onservativity and advantages of their symmetry. A wide spetrum of differene shemes based on the finite differene methods is used for numerial simulations. The numerial solving of the partial differential equations (PDEs is today used not only in suh traditional branhes, as the weather foreast or oeanography. Conneted with the tehnial development, many engineering and industrial appliations, where the numerial modeling an be suessfully used, ame to view. On the other hand, the siene provides many problems, where numerial simulations is the only approah to study them. Although the field of numerial simulations is very large, our interest fouses on the simulations of the behavior of the ompressible fluid, symbolized mathematially by a system of partial differential equations of hyperboli type. Solution of suh equations is represented as a density distribution of the onservative quantities in the omputational domain, in the partiular time of the simulation. In the ase of ompressible fluid dynamis, it is usual to neglet the fluid visosity and the full set of the Navier-Stokes equations redues to the form of system alled the Euler equations. Here, the onservative quantities beome the total mass, the total energy, and the omponents of the total momentum in eah diretion. In the ompressible fluid dynamis, the phenomena of shok waves an appear. Shok wave is a severe hange in the fluid density, veloity, pressure, and internal energy in a small distane in spae and time. The shok waves have the form of disontinuity in the solutions and an ause serious troubles to the numerial methods osillations or diffusion an appear around the shok, when numerial treatment is not adequate. In reality, the shok waves thikness is proportional to the free mean path of the fluid partiles and no real disontinuity exists. The numerial methods have to deal with suh phenomena, and alulate the solution of the Euler equations lose to the realisti fluid behavior. For plasma simulations, two approahes are possible the kineti and the fluid models. In the kineti model, the omplete Boltzmann equation is solved either by a diret or statistial method (suh as Partile in Cell or Monte Carlo method, and the solution has the form of the veloity distribution of eletrons and ions. Kineti approah may provide more omplete information and use less assumptions than fluid approah, but it is omputationally more expensive. Fluid approximation assumes that the sale lengths of intense quantities are large ompared with eletron and ion mean free paths. The temporal variations of these quantities are assumed slow ompared with ollision frequeny. For these onditions, the differene of the partile distribution from Maxwellians is negligible. As eletron ion relaxation is muh slower than distribution relaxation to Maxwellian, eletron and ion temperatures 4

7 often differ in the hot, low density orona. However, the differene is negligible in dense areas and thus we limit ourselves to one temperature approximation. Generally, these onditions an be violated in speial ases, suh as in extremely low pressure rarefied gas flows, where the fluid approah annot be applied. On the other hand, the omputational time of fluid simulations is reasonable also for large-sale simulations (when ompared with the kineti simulations, and allows simulations in higher dimensions. The first goal of this thesis is the development of an effiient, linearity and loal-bound preserving remapping algorithm for reomputation of the onservative quantities between similar omputational meshes, and its generalization to the ylindrial geometry. For future use, this algorithm must be generalizable to 3D and to meshes with hanging topology. The next goal is the development of an effiient and reasonably aurate method for the simulations of laser-matter interation and high veloity impats, based on the fluid Arbitrary Lagrangian-Eulerian (ALE methodology. This method have to treat severe omputational mesh motion and produe results omparable with the experimental data. The method must be able to reprodue the experiments in both Cartesian and ylindrial geometries, and testing of the method on a set of experiment-inspired simulations is required. This thesis desribes the onstrution of suh method, and analyzes its properties. The last main goal of this thesis is the implementation of the presented ALE method, development of the omputer ode, and performing simulations of laser-matter and high veloity impat problems to model seleted experiments. In this hapter, we summarize the main ontributions of this thesis to the field of ALE methods, and laser plasma simulations. The rest of the thesis is organized as follows. In the next hapter, we disuss two main approahes for solving the equations of ompressible hydrodynamis the Eulerian and the Lagrangian methods. These methods are suitable for different types of problems being solved. We summarize the main advantages and disadvantages of both approahes, and present the reasons for using their ombinations the Arbitrary Lagrangian-Eulerian (ALE methods. We disuss the main types of the ALE methods, and briefly go through the history of the ALE methods. Several representative Eulerian, Lagrangian, and ALE odes for ompressible fluid and plasma simulations are resumed. In hapter 3, the omplete Cartesian ALE method is presented. We go through all three parts of the ALE algorithm Lagrangian solver, mesh smoothing tehniques, and remapping. The ompatible staggered Lagrangian method based on the evaluation of several types of fores affeting the nodal movement is reviewed, inluding several types of artifiial visosity formulation. Several both lassial and modern methods for mesh smoothing and untangling are briefly summarized. Finally, the remapping method (the onservative interpolation of the onservative quantities based on the approximate swept integration approah, is presented. This tehnique, whih is the author mostly interested in, and whih is his main ontribution to the ALE methodology, is fully desribed and ompared with the lassial, omputationally expensive method based on the exat integration. The swept integration remapping algorithm is also generalized to 3D, and to the omputational meshes with hanging onnetivity. In hapter 4, the ALE method is generalized into the ylindrial oordinates. It follows the previous hapter 3, and desribes all points, where the ylindrial geometry hanges the Cartesian formulas. The Area-Weighted Differening (AWD Lagrangian sheme is desribed and it is shown, that this approah annot be used in our ALE formulation. Then, an alternative Control Volume (CV Lagrangian method is presented, satisfying all needs for suessful usage in the omplete ALE algorithm. As for mesh smoothing tehniques, no major hanges aording to the Cartesian geometry, are needed. The ylindrial formulas for our swept integration remapping method are derived in and easy-to-implement form. In the whole hapter 4, we emphasize the form of the formulas to be lose to Cartesian ones, and thus an easy generalization of the Cartesian ALE ode to the ylindrial geometry is possible. In hapter 5, the main properties of the omplete ALE method are ommented. We disuss the onservativity of all parts of the ALE algorithm, and their linearity preservation, whih in pratie indues seond order of auray. Satisfation of these important properties is demonstrated for the typial tests of fluid dynamis. In hapter 6, several physial aspets of the ompressible fluid simulations are disussed, whih arise when the ideal fluid is replaed by a laser-produed plasma. Properties of the quotidian equation of 5

8 state (QEOS are briefly summarized, bringing our simulation loser to the real plasma behavior. The implementation of the thermal ondutivity model with the lassial Spitzer-Harm heat ondutivity oeffiient is briefly haraterized, and its influene on different types of the laser plasma simulations is summarized. The proess of laser beam absorption at the ritial density is desribed, and we disuss the influene of the implementation of the laser plasma boundary onditions on the solution. Chapter 7 inludes numerial simulations of the laser-matter interations, and their omparison with the experimental data. We present three approahes for expressing the maximal laser beam intensity from the experimental data. The ode is applied to model several types of proesses. At first, we perform simulations of the interation of an intense laser beam with a massive Aluminum target, and ompare the final rater formations with the raters obtained from experiments. Next, simulations of the laser beam irradiating and ablatively aelerating a thin Aluminum dis flyer are performed. And finally, simulations of the high veloity impat of this flyer on a massive target are shown. Two ways of onstrution of the initial data for the impat simulations are ompared, average uniform values in the dis, and interpolated data from the dis aeleration simulations. All simulations are ompared with the experimental results, and demonstrated on one partiular example.. New Contributions of the Thesis The main ontributions of the thesis to the problematis of onservative interpolations, ALE methods, and omputational laser plasma hydrodynamis, are listed below: Introduing the swept region remapping method in D logially quadrilateral meshes in Cartesian and ylindrial geometries. 3D general meshes. D general meshes with hanging onnetivity. Introduing the omplete remapping algorithm (for all state quantities in ylindrial geometry. Development of the D ALE ode on logially-orthogonal omputational meshes working in both Cartesian and ylindrial geometries, appliable to the fluid and laser plasma simulations. Simulations of the small dis flyer ablative aeleration by an intense laser beam, and its impat to the massive target. The given points are new, and an be useful for both the ALE and laser plasma soieties. Parts of this thesis have been presented at the sientifi onferenes and published in the proeedings and in the speialized journals, for example Journal of Computational Physis [58], Computers and Fluids [36], or Czehoslovak Journal of Physis [55], other parts are submitted or in preparation for publiation. 6

9 Chapter Numerial Methods for Compressible Fluid Flows In this hapter, we desribe the main approahes for numerial solving of the ompressible fluid Euler equations the Eulerian and Lagrangian methods. We show the form of the system of the fluid Euler equations in both formulations and mention the basi methods for their solving, and seleted odes implementing them. We also desribe the ALE (Arbitrary Lagrangian-Eulerian methods, the ombination of both approahes. Finally, several odes for the plasma simulations are overviewed.. Eulerian Methods The family of Eulerian methods solves the system of the ompressible fluid Euler equations in the form (ρw t ρ t + div(ρw = + div(ρw + grad p = (. E t + div(w (E + p =, representing the laws of the onservation of mass, momentum in all diretions, and the total energy. Here, ρ denotes the fluid density, w = (u,v is the vetor of fluid veloities in D, E is the total energy, and p is the fluid pressure given by the equation of state p = p(ρ,ǫ, where ǫ = E/ρ w / is the speifi internal energy of the fluid. In the Eulerian model, the system of equations is disretized on the stati (in time omputational mesh. The onservative quantities (as mass, momentum, or total energy are transfered between the omputational ells in the form of the advetive flux through their edges. There exist a omprehensive theory desribing many aspets of solving the Euler equations. It desribes existene of shok waves, ontat disontinuities, and rarefation waves in the solution. Also dependene of the solution on the fluid veloity ompared with the sound speed is haraterized, and speial ases of initial onditions where the solution is analytially expressible, are identified. There exist a big number of CFD monographs about this topi, we reommend [6, 6, 93] for details. As for the numerial methods for solution of the hyperboli Euler equations, there exist lots of them with different properties. There are tehniques for analyzing the numerial methods as for their stability or order of auray. For an overview of the numerial Eulerian shemes and their analysis, see [93, 34]. Comparison of several Eulerian-type methods is done in [66]. Many Eulerian odes and simulation tools exist for modeling of different phenomena in the field of ompressible fluid dynamis, for different purposes. Let us review several of them. On the first plae, we name the pakage CLAWPACK [63] developed by Randall LeVeque at the University of Washington. This pakage for solving time dependent onservation laws in D, D, and 3D (inluding 7

10 the system of Euler equations, uses sophistiated flux splitting sheme based on advetion ideas. The ode is adapted for the user to inlude his own numerial sheme, and use it in the CLAWPACK. Parallel version of the ode is available, as well as the adaptive mesh refinement extension named AMRCLAW. ALLSPD-3D Combustor Code [] is a numerial tool developed in NASA for simulating the hemially reating flows in the aerospae propulsion systems. The ode an simulate multi-phase turbulenes and flows with wide range of Mah-number in ombustors of omplex geometry. It uses 3D multi-zone strutured grids with internal blokages and finite differene, ompressible flow formulation with low Mah number preonditioning for both laminar and turbulent flows. Another important simulations ode FLASH [5], is being developed at the University of Chiago. It is a modular, adaptive, parallel simulation ode apable of handling general ompressible flow problems in astrophysis. Its aim is to solve the long-standing problem of thermonulear flashes on the surfaes of ompat stars suh as neutron stars and white dwarf stars, and in the interior of white dwarfs. Finally, we mention the SAGE [37] (Simple Adaptive Grid Eulerian ode developed in the Lawrene Livermore national Laboratory and the Los Alamos National Laboratory. It is a massively parallel multimaterial hydrodynamial ode for solving of high deformation flow problems. By adding the temperature radiation to the ode, RAGE ode was developed. These odes are appliable for a variety of simulations, suh as laser-target interation, meteorite impat simulations, or tsunami wave spread simulations. The SAGE/RAGE odes are not available for publi usage.. Lagrangian Methods The philosophy of the Lagrangian-type methods is ompletely different from the Eulerian philosophy. In the Lagrangian model, the onservation system is solved in the form where d dt = t + x t dρ ρ dt = w ρ dw dt ρ dǫ dt = p (. = p w, x + y t y = t + w (.3 is the Lagrangian derivation in D Cartesian geometry. The Lagrangian method solves this system on the omputational mesh moving with the fluid aording to the ordinary differential equations ż n = w n (.4 for all nodes n, where z n (t = (x n (t,y n (t is the position of node n and w n its veloity. The symbol ż n denotes temporal derivative of the nodal position. Masses in all ells remain onstant. There is no mass flux (and thus no advetive momentum or energy flux through the ell edges as in the Eulerian model. For ompleteness, we demonstrate here, that the Eulerian (. and Lagrangian (. systems are equivalent. We show this equivalene for the ase of D Cartesian equations with veloity w = u. The main point is substitution of the total derivatives in the Lagrangian equations by the partial ones (.3. By this proess, the first equation hanges to ( ρ t + x t ρ + ρ x ( u =. (.5 x Here, the temporal partial derivative of the oordinate is equal to the atual fluid speed, that is ρ t + u ρ x + ρ u =, (.6 x 8

11 and after putting the derivatives together ρ t + (ρu =, (.7 x the Eulerian equation for mass onservation is derived. By doing similar proess with the momentum equation, we get ( u ρ t + x u + p t x x = (.8 and after some algebra ρ u t u + ρu x + p =. (.9 x To get the Eulerian equation for momentum, let us rewrite (.9 in the form (ρu t u ρ t + (ρu u (ρu x x + p =. (. x The two terms with minus sign orrespond to the mass equation (.7 multiplied by the veloity fator, thus they disappear and momentum equation from (. remains. By the similar proess, the energy equation an be transformed to the Eulerian methodology. For this transformation, the definition of the speifi internal energy ǫ from the previous setion, and both mass and momentum equations in Eulerian form are required. Analogial proess an be repeated to show the equality of the Lagrangian and the Eulerian formulation also in the ase of multidimensional Euler equations, and the Euler equations in the ylindrial geometry. The usual proess in the Lagrangian method is following. At first, the fores due to the pressure gradient are evaluated from the momentum equations. The fores identify the veloities in eah node defining the nodal movement. After moving the nodes aording to the fores, a new mesh appears, the internal energy is updated from the energy equation, and density is updated from new mesh and onstant ells masses Lagrangian assumption. There exist many types of the Lagrangian methods. The beginning of the Lagrangian history is in late 4s, when the Lagrangian oordinates for the fluid equations were introdued by Courant and Friedrihs [3]. The lassial methods redue the system of the Lagrangian equations to the Hamiltonian system [33] orresponding to the transformation from the motion to the ation equations, naturally preserving the total energy. The systemati approah of deriving the disrete operators with the same properties, as the ontinuous ones, has been developed by Shashkov and is presented in his book [84]. This approah is used in [8] to develop the onservative Lagrangian disretization in ylindrial oordinates. Lagrangian methods are a standard tool in CFD and are widely used for fluid simulations. Espeially, in the simulations of laser plasma, where huge hanges of the omputational domain during the simulations appear, the Lagrangian methods beome eligible due to the natural treatment of the moving boundary onditions, whih ause serious problems to the Eulerian methods. Let us overview few Lagrangian odes. The first ode is the lassial D HYDRO ode developed and used in the Los Alamos National Laboratory. This ode is based on the algorithm desribed in [83], and represents a wide range of Lagrangian hydrodynamial odes quite similar to HYDRO. Speialized Lagrangian ode for the laser-plasma interations ATLANT [43] was developed in the Russian Institute of Mathematial Modeling. This D Lagrangian ode implements the hydrodynamial onefluid model with two temperatures. Collisional laser absorption inluding the ray-traing laser beam propagation algorithm is inluded. This ode has been applied for many simulations of experiments of laser-plasma interations. By inluding the radiation transport in the multi-group approximation, ode LATRANT [4] was developed. 9

12 .3 ALE Methods Both the Eulerian and Lagrangian methods have their advantages and disadvantages, making them suitable for solving of different types of problems. The Eulerian methods are exellent for the simulations, where the omputational domain does not hange very muh, and its size and shape remains more or less the same. Eulerian methods are general, appliable to solving of arbitrary system of hyperboli equations, robust, and stable. On the other hand, the Lagrangian methods are exellent for solving of problems, where signifiant hanges of the omputational domain appear, or for problems of ontats of different materials. The hanging Lagrangian mesh naturally overs the whole omputational domain, the material interfae is naturally kept sharp and in right diretion, whih needs additional ompliated proedures in the Eulerian methods. On the other hand, several problems may appear in the Lagrangian proess due to the mesh movement. The omputational mesh an degenerate ells with a very small volume an appear, opposite ell edges an interset eah other (hourglass-type ell motion, and negative volume ells an arise. In suh situation, the Lagrangian method either fails, or (if treated by the method quikly inreases its numerial error. One way, how to deal with these problems, is to use the Arbitrary Lagrangian-Eulerian (ALE methods (for more details, see hapter 3. The ALE algorithm was firstly proposed by Hirt in 974 [4] and at the beginning, it was used often for the simulations of the solid body deformations. Many authors ontributed to this topi [7, 9,,, 79, 6, 8], but the CFD soiety did not pay enough attention to them. During the last several years, the situation has hanged, and the ALE methods are beoming a modern tool for fluid simulations [47, 4, 3]. When talking about the ALE methods, one an distinguish two main types of the methods. In the first type, the movement of the omputational mesh is predefined, using some information about the problem solution. Thus, no tangling of the mesh an appear [8]. In the seond type (preferred in our approah [79, 4, 3], the purely Lagrangian movement is performed. In the situation, when the mesh degenerates, or its quality is low, or just after eah several Lagrangian steps, the mesh smoothing tehnique is applied and followed by the onservative reomputing (remapping of the onservative quantities to the smoother mesh. Both types of ALE methods are used for real omputations in modern ALE odes. Let us briefly summarize several known ALE odes, used for the CFD simulations. Probably the first ALE ode developed and appliable for pratial usage is YAQUI [], implementing the original ALE algorithm [4] by its authors. The programs omputational sheme was later improved a little, and ode SALE-D [] was released. These odes solve the Navier-Stokes equations on D omputational mesh, whih an either move with the fluid in a typial Lagrangian fashion, an be held fixed in an Eulerian manner, or an move in some arbitrarily speified way. 3D version of the ode was also developed. Another and muh newer ALE ode is alled ICF3D-Hydro [88], and as lear from the name, it is a 3D ALE ode for simulations of inertial onfinement fusion plasmas. It handles general 3D unstrutured meshes, using a seond order FEM Godunov sheme with a 3D generalization of van Leer slope limiter proess for shok stabilization. The ode an work in a fully parallel mode on both SMP and MPP arhitetures. Another ICF oriented ode is DRACO [46], whih is still under development. It is the ALE hydrodynamial ode designated to run in D, D, and 3D in planar, spherial, and ylindrial geometries. The ode uses Lagrangian hydro step, seond order rezoning algorithm, and interfae traking method. The ode uses SESAME equation of state [], and works in parallel. For simulations of the fluid behavior involving strong shok waves, the ALEGRA [79] (ALE General Researh Appliation ode was developed in the Sandia National Laboratory. It uses the ALE algorithm on an unstrutured mesh, ombining the properties of modern fluid Eulerian shok odes with Lagrangian solid mehanis odes. Moreover, magnetohydrodynamis (MHD is inluded into it, allowing to perform high energy density plasma simulations (as z-pinh simulations. Another sophistiated ALE ode CALE [9] was developed in the Lawrene Livermore National Laboratory, and uses a staggered D quadrilateral strutured mesh Lagrange sheme followed by the mesh smoothing and remapping step. The ode inludes a multi-group, flux-limited, radiation diffusion model

13 in a wide variety of plasma models. The extension of the ode CALEICF inludes the thermonulear burning mehanism, and is also extended to 3D HYDRA [73] ode representing the state of the art of 3D ICF odes. One of the top ALE odes is the ode CORVUS [6] developed at the British Atomi Weapons Establishment (AWE. This robust and aurate multimaterial ode works on the general unstrutured D meshes. The main strength of the ode is in its treatment of the material interfaes. Today, the ode is used in different areas, suh as the simulations of highly explosive materials explosions, shok/bubble interations, or high veloity projetile impat to the tank of water. The ode is urrently being extended to the 3D fully parallel ode PEGASUS. Finally, we refer to the ALE ode ALE In. [69], being urrently under development in the Los Alamos National Laboratory. This D ode uses exatly the same approah, as this thesis Lagrangian step based on the ompatible disretization [84, 8], several types of mesh rezoning tehniques, and sweptregion integration based remapping method [58]. The author of this thesis ommuniates with the authors of the ALE In. ode a lot, ollaborate in several fields, and many ideas and algorithms are similar. The main differene is in the purpose of the odes. The ALE In. ode is the inubator the testing environment for onfirming of the properties of the developed methods in different aspets of the ALE field (a similar developmental ode fousing on the ombination of the ALE method with the adaptive mesh refinement approah (AMR is desribed in [3]. On the other hand, our ode desribed in this thesis, is a speialized ALE ode for the laser plasma simulations, inluding many physial aspets of the simulations, suh as heat ondutivity, plasma equation of state, or the interation of plasma with a laser beam..4 Hydrodynamial Codes for Plasma Simulations The field of plasma physis appeared at the turn of the 9th and th entury. The refletion of radio waves led to the disovery of the Earth s ionosphere, a layer of partially ionized gas in the upper atmosphere. During the first half of the th entury, many astrophysial disoveries produed a requirement of understanding the plasma physis. Around the year 94, Hannes Alfvén formulated the theory of magnetohydrodynamis (MHD, in whih plasma is treated as a onduting fluid, and its onlusions were onfirmed by astrophysial observations. The reation of the hydrogen bomb in 95 generated a great deal of interest in ontrolled thermonulear fusion as a possible power soure for the future in the late 95 s and the early 96 s. In these years, theoretial plasma physis first emerged as a mathematially rigorous disipline. The development of high powered lasers in the 96 s opened up the field of laser plasma physis. When an intense laser beam strikes a solid target, material is immediately ablated, and a plasma forms at the boundary between the beam and the target. The major appliation of laser plasma physis is the approah to fusion energy known as the inertial onfinement fusion. The requirement of the simulation tools for plasma physis arises, leading to the development of many odes for numerial modeling of various plasma physis phenomena, based on both kineti and hydrodynami approahes. In this setion, we summarize several hydrodynamial odes for the plasma simulations. The state of the art of the plasma simulation is represented by 3D odes applied in the fusion researh, typially used on high performane servers and lusters for simulations of the inertial onfinement fusion (ICF proesses. Suh odes usually solve the full set of the Maxwell equations by their diret integration, taking all eletromagneti effets into aount, suh as harges and urrents generated by the harges in the plasma, or the onnetion of the full spetrum radiation and the plasma harges. In suh ritial ICF simulations, proesses as the ion-ion and ion-wall ollisions has to be taken into aount. Suh sophistiated and omplex odes do not only simulate performed experiments, but are also used for investigating new and unknown phenomena in the plasma field. In the previous three setions, we have already desribed several omplex odes used for plasma simulations, Eulerian FLASH and SAGE/RAGE odes, the Lagrangian ATLANT ode, and ALE ICF3D-Hydro, DRACO, ALEGRA, and CALE odes. All of them were based on the fluid models of plasma. Let us review

14 several more odes for fluid plasma simulations MEDUSA and ALPS, and the hybrid fluid/partile odes LASNEX and VORPAL. Classial ode MEDUSA [] is a D Lagrangian hydrodynamial ode, whih was written to investigate some of the hydrodynami and plasma proesses that take plae in a small pellet whih is irradiated by laser light. Its main purpose is to perform simulations of proesses appearing in ICF, and verify its feasibility. In the ode s model, the Navier-Stokes equations are supplemented by separate heat ondution equations for the ion and eletron temperatures, and a variety of additional effets are inluded. Adaptive Laser Plasma Simulator (ALPS [9] developed in the Lawrene Livermore National Laboratory is a D and 3D hydrodynamial ode, using an Eulerian high-order aurate upwind tehnique ombined with the adaptive mesh refinement (AMR approah. The laser beam propagation is modeled using a paraxial wave equation. The main goal of this ode is in modeling of laser plasma instabilities, and developing tehniques to make laser beams more resistant to them. The LASNEX ode [4] was developed in the Lawrene Livermore National Laboratory to study inertial onfinement fusion (ICF proesses, and to onstrut and analyze ICF experiments. It was first referred to in literature in 97, but over time the ode has evolved and has been greatly enhaned. It inludes the Lagrangian (or simple ALE hydrodynamis, the eletron, ion, and radiation heat ondution, and the oupling among these energy fields. Thermonulear reation an be modeled by LASNEX, inluding the energy produed as well as the reation produts. It provides suffiient agreement between the alulations and the experiments, whih makes this ode eligible to predit and design future experiments, suh as on the National Ignition Faility, for example. Finally, we state the ode developed at the University of Colorado VORPAL [76], a versatile plasma simulations ode. It inludes both PIC and fluid plasma models, inorporating several models of plasma and eletromagneti field behavior in D, D, and 3D. Both modes (PIC and fluid an run independently or in a hybrid regime. VORPAL is urrently being used to study a number of plasma physis problems, suh as generation of laser wake fields, optial injetion of partiles into wake fields, or radio frequeny heating of fusion plasmas.

15 Chapter 3 ALE Method in Cartesian Geometry In this setion, we desribe the basi priniples of the ALE methods in the Cartesian geometry. We go through all three main parts of the ALE algorithm the Lagrangian solver, mesh smoothing tehniques, and onservative interpolation algorithms. We fous on the remapping tehniques, whih are the main ontribution of the author to the field of ALE methods. The ALE algorithm is based on the ombination of the Eulerian and the Lagrangian methods. The omputational mesh moves with the fluid (as in the Lagrangian approah, however the mass flux over ell boundaries is allowed (as in the Eulerian approah. The ALE method ombines advantages of both approahes. Beause of the moving mesh, it allows to simulate problems on moving or hanging omputational domain. The omputational mesh is smoothed from time to time, ausing less problems with negative ell volumes, hourglass-type motion, sharp edge angles, and other onfigurations, whih are hardly to treat by pure Lagrangian methods. Initialization t = i = Main Program Loop Compute Timestep dt t = t + dt i = i + i = Perform Lagrangian Step if (i>i yes max or (bad_quality_mesh no Perform Mesh Smoothing Perform Remapping of All Quantities to the New Mesh Until t<t max End Figure 3.: Struture of the ALE algorithm. In the ALE algorithm, the Lagrangian-type and Eulerian-type steps are used. At first, several time steps of the Lagrangian solver are performed. After a seleted number of suh steps, or if the mesh beomes invalid or has bad quality, the Eulerian-type step is used. It inludes a mesh smoothing sub- 3

16 step providing us a new mesh, whih is already regular and better quality, but similar to the original Lagrangian mesh. The seond part of the Eulerian-type step is the onservative interpolation substep, whih onservatively remaps all onservative quantities from the Lagrangian mesh, to the new, smoothed one. The basi struture of the ALE algorithm is shown in Figure 3.. Let us go through all the mentioned parts of the ALE method in detail. 3. Lagrangian Solver There exist many approahes for formulating the Lagrangian shemes. For ompleteness we mention here the well known point-entered method with all quantities defined in the enters of the omputational ells, reviewed in [8]. In general, this type of methods has problems with the total energy onservation. On the other hand, it is quite easy to inorporate suh method into omplete ALE ode. We employ the lassial staggered Lagrangian methods with the salar quantities (suh as fluid pressure, density, or internal energy defined in the enters of the omputational mesh ells, and the vetor quantities (suh as positions or veloities defined in the mesh nodes. We use disretization from [8, 9,, 6], whih is one of the most suitable approahes for the Lagrangian hydrodynamial simulations of the fluid dynamis. This method is fully onservative, and allows a natural treatment of boundary onditions, whih are very important for laser plasma simulations. On the other hand, due to the staggered disretization, it requires more sophistiated method for onservative quantity remapping to make the Lagrangian solver ompatible with the rest of the ALE ode. The Lagrangian solver is based on the omputation of three types of fores in eah node of the omputational mesh, and movement of the nodes aording to these fores. They inlude the zonal pressure fore, the subzonal pressure fore, and the artifiial visosity fore. The zonal pressure fore represents the total fore of the fluid aused by the pressure gradient, affeting the node from all ells around it. The subzonal pressure fore arises from the finer disretization (subzones, and prevents the nodes from unphysial hourglass-type motion. The visosity fore adds artifiial diffusion to the solution, whih makes the Lagrangian solver able to perform also simulations inluding shok waves or ontat disontinuities. Let us disuss the Lagrangian solver and the fores in more details. 3.. Geometrial Struture The Lagrangian solver uses staggered subzonal disretization on logially-orthogonal omputational mesh haraterized in Figure 3.. Eah ell is divided into four subzones, separated by four dashed lines (separators onneting the ell enter with the enter of eah edge of the ell. All ell edges, subzones, and dashed separators are enumerated from to 4 in the partiular ell, starting from the lower (edges, separators or lower-left (subzones ones, in the ounter-lok wise order. For further integration, we also need orientation of all ell edges and separators, so let us define, that all separators s point from the ell enter to the orresponding edge enter, and all subzonal edges a +, a have their diretions from the nodes to the edge enters. The enters of the edges are omputed as the average of the oordinates of both end nodes. The original Lagrangian sheme uses simple averaging also for the omputation of the ell enters. To inorporate the Lagrangian method into the ALE ontext, we hanged the ell enters to enters of mass (entroids, whih is needed by the remapping part of the algorithm. This hange does not affet the order of auray of the Lagrangian sheme, as demonstrated later. The entroid of ell is omputed as x = xdxdy V, y = y dxdy V, (3. where the formula for the volume of ell is V = dxdy. (3. 4

17 a + 4 a i,j 4 4 i,j a 4 s a 4 s 3 3 i,j s + a + 3 s a 3 i,j a i,j a + 3 Figure 3.: Struture of one omputational ell i,j and enumeration of the nodes around it. It inludes enumeration of ell edges and ell subzones, and orientation of a l, a+ l, and s l vetors, l =,...,4. By using the Green formula, the volume integrals an be redued to the boundary integrals V = xdy (3.3 x = y = e E( e e E( e x dy (3.4 V xy dy e E( e V, (3.5 where E( is set of edges of ell. After evaluating the line integrals, we get the final formula V = ((y y (x + x + (y 3 y (x + x 3 + (y 4 y 3 (x 3 + x 4 + (y y 4 (x 4 + x (3.6 for the volume of any quadrilateral ell with nodal oordinates (x,y, (x,y, (x 3,y 3, (x 4,y 4 in ounter-lok wise order. Analogially, we get the formula for the oordinates of the entroid of the omputational ell. It an be shown that the final formulas are the same, if the opposite part of the Green formula is used, for example if the ell volume is omputed as V = y dx. (3.7 e E( This is, of ourse, expeted, but it gives us a simple test of orretness of our formulas. Subzonal mass of subzone l =... 4 of ell is naturally expressed as e m l = ρl V l, (3.8 where the subzonal volume V l is omputed using the same formula as for the zonal volume (3.6, ρ l represents subzonal density defined by formula (3.8, and the subzonal mass ml results from the initialization proess and remains unhanged during the whole Lagrangian omputation. Nodal and ell 5

18 (zonal masses are naturally defined as m n = C(n m ln, m = 4 m l (3.9 respetively. Here l n means subzone of ell orresponding to node n, and C(n is a set of all ells inluding node n. This allows us to define the zonal (ell density (and also nodal density similarly naturally as ρ = m, V (3. beause 3.. Zonal Pressure Fore V = By integration of the momentum equation in x diretion 4 l= ρ u t = p x l= V l. (3. (3. over the nodal volume (four subzones from four different ells around a ommon node haraterized in Figure 3.3, we get ( u p m n = dxdy, (3.3 t n V n x where m n is the nodal mass (3.9, V n is the nodal volume defined as sum of the orresponding subzonal volumes, and ( u/ t n is the average temporal derivative of the fluid veloity in V n. We have assumed that the nodal veloity u n is onstant in the whole nodal volume V n. The right hand side of equation (3.3 represents the total fore affeting the nodal movement in x diretion, whih we denote by the symbol Fn. x By applying the Green theorem to the right hand side, we get the boundary integral Fn x = p dy, (3.4 V n whih an be deomposed into a sum of edge integrals ( 4 Fn x = l= l= s (n,l l p dy + p dy s (n,l l, (3.5 where by the symbol (n, l we mean the ell, whose subzone with the index l orresponds to the node n, and s l denotes the separator inside ell with the index l. In this thesis, the l index is yli (as modulo of 4, in the sense that s = s 4, and similarly for all terms, where l appears. Here, we assume onstant pressure p inside eah ell. Then, one an put pressures in front of the integrals and rewrite the formula as ( 4 Fn x = p (n,l dy dy. (3.6 s (n,l l s (n,l l The edge integrals an be evaluated from the ending points of the separators, so let us define ( I s (n,l l = dy = y (n,l y (n,l e, (3.7 l s (n,l l Generally, in this paper, we use apitals to note sets of objets, for example C( means all ells in the neighborhood of ell (in our logially retangular mesh, C( is a 3 3 path of ells sharing an edge or vertex with, inluding ell, E(n are all edges onneted to node n, or N( represents all nodes of ell. 6

19 i,j+ i,j+ i+,j+ i,j+ i,j+ s i,j+ s a a+ i+,j+ 4 s i+,j+ i+,j+ s i,j a+ a 3 i,j 4 i,j a 3 4 a+ s3 + a s 3 i,j a4 i,j s 3 i+,j 4 s i+,j i+,j i+,j i,j i,j i+,j Figure 3.3: Situation in a nodal volume orresponding to node i,j and four ells around: i,j, i +,j, i +,j +, and i,j +. Outer normals of ell sides (vetors a ± l and normals of separators (vetors s l are shematially shown. where the symbol e (n,l l represents the edge of ell (n,l with index l, and the formulas for the oordinate of the ell enter y and edge enter y e are defined in the previous setion 3... Now, one an rewrite the formula as 4 ( 4 Fn x = p (n,l I(s (n,l l I(s (n,l l = Fp x (n,l l, (3.8 l= or expliitly for the node n = i, j as F x i,j = p i+,j+ ( I(s i+,j+ I(s i+,j+ 4 + p i,j+ ( l= I(s i,j+. I(s i,j+ ( ( + p i,j I(s i,j 3 I(si,j + p i+,j I(s i+,j 4 I(s i+,j 3 (3.9 Every term of the previous formula orresponds to the orner pressure fore Fp l (n,l, influening the node from the partiular subzone. In the later text, we will denote the nodal pressure fore by the symbol Fp x n. For ompleteness, exatly the same formula an be used for the omputation of the zonal pressure fore in y diretion F y n = 4 ( p (n,l l= J(s (n,l l J(s (n,l l = 7 4 l= Fp y l (n,l, (3.

20 just the integrals are omputed as ( J s (n,l l = s (n,l l dx = x (n,l + x e (n,l l, (3. the different sign arises from the Green theorem. These relations give us the omplete formulas for the omputation of the zonal fores from eah subzone to the partiular node Subzonal Pressure Fore To explain the subzonal pressure fore onept, let us suppose now that the pressure is not onstant in eah ell. Let us make finer disretization let us define the subzonal pressures as or the subzonal pressure differene p l = v γ ρl (3. δp l = v γ (ρl ρ, (3.3 where the sound speed v is omputed from the equation of state, and γ is the ratio of the speifi heats. The zonal and subzonal densities are omputed by dividing the partiular mass by the orresponding volume. These zonal and subzonal masses are equal to the initial values from the first Lagrangian step, the densities hange in time as the omputational mesh moves. Easily, the following relation holds p l = p + δp l, (3.4 whih expresses the subzonal pressure as the variation of the zonal one. Now, let us repeat the same proess, as in the previous setion. It ontinues till the formula (3.5 exatly the same way, as before, but now, one annot put the pressure in front of the integrals, beause there is no unique pressure value in the ell. Thus, one have to define a value of pressure along the separator lines, whih an simply be done by averaging the adjaent subzonal pressures. Then, the formula an be rewritten as l= F x n = 4 l= ( pl (n,l + pl+ (n,l s (n,l l dy + pl (n,l + pl (n,l s (n,l l and after putting the formula (3.4 and the notation (3.7, we get ( 4 Fn x = (p (n,l + δpl (n,l + δpl+ (n,l I(s (n,l l + (p (n,l + δpl (n,l + δpl (n,l dy, (3.5 I(s (n,l l (3.6 After regrouping the terms, one an write the formula in several different forms inluding the zonal pressure fore Fp x n defined in (3.8, F x n = Fp x n + = Fp x n + = Fp x n l= 4 l= 4 l= (( δp l (n,l + δpl+ (n,l δp l (n,l 4 l= ( I(s (n,l l I(s (n,l l ( I(s (n,l l δp l+ (n,l δpl (n,l + Fdp x l (n,l. ( I(s (n,l l δp l (n,l + δpl (n,l I(s (n,l l 4 l= ( I(s (n,l l δp l (n,l δpl (n,l. (3.7 (3.8 8

21 Espeially, the formulation (3.8 is important, beause it demonstrates the geometrial meaning of the subzonal pressure fore. The first term of the fore orrets diretly the zonal pressure fore due to the different pressure in the zone and in the subzone. The seond and the third terms are assoiated with the midpoints of the partiular ell edges and play a very important role in reduing the artifiial mesh movement. For more details about the subzonal disretization and different fore formulations, see [9]. Let us also note, that (exatly as in the previous setion 3.. the subzonal pressure fore in y diretion an be omputed using the same formulas, just the integrals must be defined as in (3.. Due to the fat, that the integral of over the boundary of a subzone must be equal to zero (subzone is losed = V l dy = I(s l + I(s l I(a l + I(a + l, (3.9 the integrals over the separators s an be replaed by the integrals over the outer half-edges of the ell a, I(s l I(s l = I(a+ l I(a l, (3.3 where I(a ± l = dy = y e(±,l y n(,l. (3.3 a ± l This allows us to reformulate the fore formulas (3.8, (3.8, whih may be useful in the partiular implementations. In the later text, we use the term subzonal pressure fore and notation Fdp x n for the part of the formula (3.8 not inluding the original zonal pressure fore Fp x n. In the real omputations, we multiply this subzonal pressure fore by the merit fator f M,. It is an important issue for setting the amount of the subzonal pressure fore, whih regulates the measure of removing the unreal hourglass-type motion of the omputational mesh. It is not generally easy to say, how to selet the merit fator, in real simulations we usually selet it lose to the enter of the feasible interval, f M /. Automatization of seleting the merit fator is analyzed in [9] Visosity Fore Visosity is an important part of the total nodal fore. Without the artifiial visosity, the Lagrangian solver is not able to simulate problems inluding shok waves and ontat disontinuities. This is important partiularly in the field of laser plasma target impats, whih we fous on. There exist many approahes, how to inorporate the artifiial visosity to the solution. The original approah oming from [95] adds a nonlinear term in the form q = C ρ ( w (3.3 to the pressure p in ell, where shok ompression appears. Here, C is an unity order onstant, and w is the veloity differene over this ell. Purpose of this term is preventing the ell to ollapse during the omputational time step t satisfying the CFL stability ondition v t/ x, where v is the sound speed in ell appropriate to the fluid pressure p + q, and x is the harateristi length of the zone. To enfore the visosity pressure to be dissipative, we define it non-zero only in the ase of zonal ompression. Then, the nodal veloity (and onsequently kineti energy is via this mehanism transformed into the higher pressure (and onsequently into the higher internal energy, so it ats as the real visosity. The previous formulation prevents the zone from ollapsing, but does not remove the unphysial osillations around the shok wave. To eliminate them, linear visosity formulation was developed [6] q = C ρ v w, (3.33 9

22 whih diminish slower around the shok wave and thus eliminates more ompression in the ontiguous ells. Many Lagrangian odes use the artifiial visosity in the form of ombination of the previous two formulas, q = C ρ v w + C ρ ( w, (3.34 and typially the following relation for the onstants holds, C C. Finally, we present here the Kuropatenko ombination of the linear and non-linear visosity ombination [59, 96], whih we will denote by q Kur : q Kur = ρ C γ + 4 w + C ( γ + ( w + C 4 (v w. (3.35 Typially, the oeffiients are seleted as C = C =. Obviously, this formula redues to the linear visosity for v, and to the non-linear formula for v. Both in our pratial tests and [], this formulation produes high-quality results and we use it as a visosity pressure for our simulations. However, there are several ways how to inorporate the visosity pressure into the visosity nodal fores whih we desribe later. Bulk Visosity The easiest way to add the visosity to the solution, is the bulk formulation []. The first possibility is to inlude the Kuropatenko visosity pressure (3.35 diretly into the pressure of eah zone, and ompute the zonal pressure fores Fp x n (3.8 in eah node n from suh modified pressures p + q Kur. The seond way is to onstrut the subzonal visosity fores, and add them to the total nodal fore. In fat, this approah is not needed in the ase of bulk visosity, but it is required for other visosity formulations, where the easy way is not possible. In the ase of the bulk visosity, the visosity fore is Fq x n = 4 l= ( q(n,l Kur I(s (n,l l I(s (n,l l = 4 l= Fq x l (n,l (3.36 and analogially in the y diretion, just the pressure is replaed by the Kuropatenko visosity pressure in equation (3.8. The equivalene of both approahes is apparent. Edge Visosity The edge visosity is also onstruted to derease the kineti energy and transform it to the internal one. It was introdued in []. The basi priniple of the edge-entered visosity is the evaluation of the Kuropatenko visosity (3.35 along every edge e(,l of ell, q Kur e(,l = ρ e(,l C γ + 4 w e(,l + C ( γ + ( w e(,l + C 4 (v e(,l w e(,l. (3.37 Here the edge density and sound speed is omputed from the nodal densities and sound speeds on both sides of the edge ρ e(,l = ρ n(e(,l, ρ n(e(,l,+, ve(,l ρ n(e(,l, + ρ = min(v n(e(,l,,v n(e(,l,+, (3.38 n(e(,l,+ where by the symbols n(e, and n(e,+ represent the starting and ending nodes of edge e, the nodal density ρ n is naturally omputed as ρ ln V ln ρ n = C(n C(n V ln. (3.39

23 The nodal sound speed v n is obtained by the same formula, with the zonal volumes and sound speed instead of subzonal volumes and densities. The veloity differene w e along the edge e is omputed as where This allows us to write the edge-entered visosity fore as w e = δu e + δv e, (3.4 δu e = u n(e,+ u n(e, (3.4 δv e = v n(e,+ v n(e,. (3.4 fe(,l x = ( ψ e(,lqe(,l Kur δu e(,l I (s l + δv e(,l J (s l e(,l we(,l, (3.43 f y e(,l = ( ψ e(,lqe(,l Kur δu e(,l I (s l + δv e(,l J (s l e(,l we(,l. (3.44 The previous formulas are used only in the ase of ell ompression, whih means, that the following relation is true δu e(,l I (s l + δv e(,l J (s l, (3.45 otherwise the edge fores are set to zero. Eah edge-entered visosity fore is then added/subtrated to/from the total nodal visosity fore, 4 4 Fqn x = (fe((n,l,l x fx e((n,l,l = Fq(n,l x l (3.46 l= and similarly in the y diretion. We still did not omment the edge visosity limiter ψ e(,l. It helps to redue the artifiial visosity also in regions of linear veloity gradients, where the visosity is not required (ase of shok-less ompression. On the other hand, in some simulations from the field of laser plasma, the limiting proess auses less diffusion and unsteady solution, so we kept the possibility in the ode to swith the limiting off. It is neessary to test for the partiular problem type, whether to use limiting or not. As for the formulas for the limiter evaluation, we refer to the original paper [], inluding also more details about the omplete edge visosity proess. Tensor Visosity The tensor form of the artifiial visosity generalizes the previous edge formulation. Again, it is based on the salar Kuropatenko visosity term, multiplied by the tensor of veloity gradient. Analogially as in the edge visosity ase, the tensor visosity an inlude a limiter term swithing the visosity off for the ase of shok-less ompression. The advantage of this formulation is the redution of dependene of the solution on the omputational mesh. It keeps the useful features of the edge visosity, but it follows the real physial visosity in a tensor form. The tensor visosity has the following form l= Q = µg, (3.47 where µ is a salar visosity oeffiient, and G = gradw is the tensor of veloity gradient. In [7], the tensor visosity formulas are derived both in ontinuous and disrete ases. When ompared with the edge visosity, the visosity fores are not edge entered any more, but they diretly affet the

24 omputational mesh node from a partiular ell subzone. In eah subzone l of ell we need to ompute the following visosity oeffiient ( µ l = ( ψρ l l γ + γ + C w l 4 + C ( w 4 l + C (v L l (3.48 based on the Kuropatenko modulus again. Here, ψ l is the visosity limiter, this time omputed in eah subzone, w l is the subzonal veloity jump omputed as the maximal veloity differene aross the orner volume, and the harateristi subell length L l is required to keep the orret magnitude of the visosity oeffiient. There are many ways to selet the harateristi length, we use the formula suggested in [7] L l V l x x = for x w avg > x w avg (3.49 V l x x for x w avg x w avg, where the average subzonal veloity w avg is omputed as the arithmetial average of the four veloity values in all subzone verties. The vetors x and x start in the edge enters of the subzone orresponding to the ell edges, and point to the enters of the separators on the opposite side of the subzone. The final ompliated formula for the visosity fore Fq l from eah subzone to the orresponding node inluding the salar visosity oeffiient µ l is presented in [7]. As in the ase of the other visosity types, just by summing all these fores around a node, we get the total nodal visosity fore Conservative Internal Energy Update In setions 3.. and 3..3 we have shown, that the fore is evaluated by integration of the momentum equation. Simply, by substituting the temporal veloity derivative by the entral differene ( ( w w m n = m n w n n = F n, (3.5 t t n and just by moving the old veloity w n to the right hand side of the equation, we get the formula for the new veloity w n omputation w n = w n + t m n F n. (3.5 Thus, the momentum onservation in eah ell is satisfied due to the fore definition and the loseness of the ell. Now, let us derive the formula for the new speifi internal energy omputation. At first, we have to define the total energy in ell as E = m ǫ + 4 l= ( ml u n(,l + v n(,l, (3.5 whih orresponds well to the original formula for the total energy density defined in setion.. Due to the onservation of total energy in eah ell, the time derivative of this quantity is equal to zero and an be written as = E t ǫ 4 ( = m t + u n(,l v n(,l ml u n(,l + v t n(,l t l= and due to equation (3.5, one an rewrite it as = m ǫ t + 4 l= ( u n(,l F x l ( v n(,l F y l. (3.54

25 Now, let us define the total work, performed in ell due to the fores F as E work = whih allows us to rewrite the previous equation in the form 4 w n(,l F l, (3.55 l= m ǫ t = Ework. (3.56 By substituting the speifi internal energy derivative by the entral differene, we get the final formula for the new speifi internal energy ǫ = ǫ + t E work, (3.57 m whih is fully onservative. We have shown, that the onservation of momentum and total energy are naturally satisfied by definition of the orner fores. There is still the issue of the mass onservation. This is guaranteed by the Lagrangian nature of the step. The zonal, subzonal, and onsequently the nodal masses are omputed during the initialization stage, and then kept unhanged during the whole pure Lagrangian simulation. Only the zonal and subzonal densities are updated in eah Lagrangian step aording to the nodal movement hanging the volumes, the total mass is onserved without any ompliations. Thus, we have ompletely onservative Lagrangian sheme for fluid simulations Complete Algorithm of the Lagrangian Method In this setion, we desribe the omplete algorithm of the Lagrangian step, allowing us to solve the system of fluid equations in Lagrangian form (.. It follows the proess presented in [8, 9, 6]. When the Lagrangian step starts, the following quantities are known from the previous time step or initialization: time step t, nodal oordinates z n = (x n,y n, nodal speeds w n = (u n,v n, ell and subzone volumes V, V n, ell and subzone densities ρ, ρ n, ell pressures p, and speifi internal energies ǫ. The omplete algorithm an be desribed in the following way. For eah subzone, ompute the zonal, subzonal, and visosity pressure fores Fp l, Fdpl, and Fql desribed in setions 3.., 3..3, and 3..4 respetively.. Compute total orner and total nodal fores in eah node F l = Fp l + Fdp l + Fq l ( F n = F l (n,l (3.59 l= as a sum of all fores of all adjaent subzones. 3. Aording to the nodal fores, ompute the new veloities w n = w n + t m n F n (3.6 and apply the veloity boundary onditions to them, and ompute the veloities in the half-time w / n = ( wn + wn. (3.6 3

26 4. Move the nodes to their new positions z n = z n + t w / n (3.6 aording to the veloities in half-time. 5. Update the geometry aording to the new nodal position ompute new zonal and subzonal volumes, enters, and edge enters. 6. Compute the total work done in eah ell due to the fores affeting its nodes E work = and ompute the new speifi internal energy due this work 4 l= 7. Update ell and subzone densities in new ells as F n(,l w / n(,l (3.63 ǫ = ǫ + t m E work. (3.64 ρ = m, ρ l = ml V V l, (3.65 and new ell pressure from the equation of state, and apply the pressure boundary onditions to them. 8. Finally, swith veloities to the new mesh w n = w n. In fat, this algorithm uses the Euler method with step t for time integration, whih is fast and easy to implement, but only first order aurate. To improve the algorithm, and due to the fat, that we wish all quantities entering the definition of total energy to be defined at the same time level, better time integration sheme should be used. We use the seond-order Runge-Kutta method (RK. The modifiation is quite easy at the end of the proess, the new quantities (positions z n and the geometrial quantities as volumes, enters,..., energies, zonal and subzonal densities are transformed to the halftime t/ by averaging the omputed values with the original ones, and the desribed proess is repeated one again to get the new values in the final time t. For more details about the time integration issues, see [8]. 3. Mesh Smoothing Tehniques The seond essential part of the ALE algorithm is the method for mesh smoothing and regularization. It regularizes (untangles and smoothes the omputational mesh and produes the new one, used for further alulation. There exist many types of mesh regularization tehniques, due to our needs we fous only on the methods smoothing the meshes by node repositioning. There are several approahes for this task methods based on plain or weighted averaging, or loal and global optimization methods. Let us go through several methods for mesh optimization and disuss its properties. 3.. Plain Averaging and Winslow Smoothing The easiest methods for omputational mesh regularization are based on some kind of average alulations. The method based on the plain average alulation an ompute the new mesh nodes positions in the following way z i,j = 8 (4z i,j + z i,j + z i+,j + z i,j + z i,j+. (3.66 4

27 where z i,j = (x i,j,y i,j denotes the nodal position of the node n = (i,j, and the tilde symbol denotes the quantities onneted to the new smoothed omputational mesh. This simple formula is fast and smoothes the mesh properly. On the other hand, the mesh motion an be too severe, and the new mesh an be too different from the original one. Another method using priniples of averaging for mesh smoothing, is the lassial Winslow smoothing introdued in [97] and extended in [9] and [5]. This smoothing sheme results from the solution of the inverted Laplae equation and an be written in the following form ( z i,j = α (z i,j+ + z i,j + γ (z i+,j + z i,j (α + γ β (z i+,j+ z i,j+ + z i,j z i+,j, where the oeffiients α, β, γ are omputed as α = x ξ + y ξ (3.67 β = x ξ x η + y ξ y η (3.68 γ = x η + y η. (3.69 The derivatives of the nodal positions z ξ, z η aording to the logial oordinates ξ, η of node n = (i,j are omputed as z ξ = (z i+,j z i,j (3.7 z η = (z i,j+ z i,j. (3.7 This movement attempts to preserve mesh orthogonality and is used for mesh smoothing in many ALE odes [79]. Although, this method an produe meshes very different from the original Lagrangian meshes, on the other hand (unlike the plain averaging, the obtained results are reasonable and an be used for real ALE simulations. There exist some speial ases, where this method is not advisable and more sophistiated method is neessary, but in general, the Winslow smoothing method an be employed to most ALE simulations. 3.. Advaned Rezoning Methods There exist many other methods for mesh regularization and smoothing, here we briefly desribe two types of them losely onneted to the ALE approah the methods based on the ombination of global and loal mesh smoothing tehniques, and the methods based on the Referene Jaobian smoothing. Generally, the mesh smoothing tehniques an be divided into two groups global and loal optimization methods. We fous here on the loal feasible set method and the global method of the numerial optimization of mesh quality funtional, and show their ombination introdued in [94], whih an be used for untangling of hardly distorted omputational meshes. The feasible set method is a loal method moving the nodal position just with respet to the loal neighborhood of the node. Its biggest advantage is the fat, that it moves only the nodes, that are neessary to be moved. On the other hand, it is not guaranteed for a loally hardly distorted meshes that the loal method will help. For eah mesh node, its feasible set is onstruted [78], whih is the polygon, into whih one an move the node to loally orret the problemati node and make all ells around the node valid (with positive subzonal volumes. The numerial funtional optimization method is a global method moving all the mesh nodes at one. The main advantage is produing a valid mesh. On the other hand it moves all mesh nodes, also in smooth mesh regions, and thus inreases the numerial error of the oming remapping stage. At first, a funtional desribing mesh quality must be onstruted (for examples see [94], whih is then minimized by some optimization method, suh as onjugate gradient method [77]. The ombination 5

28 of both methods, introdued in [94], ombines positives of both approahes. At first, the feasible set method loally orreting most of the problemati nodes is applied. The following global numerial optimization orrets all problems, but zero-volume ells an appear. Due to the previous feasible set step, the global mesh movement in the seond step is not so strong. Then, the third step is applied loal feasible set method again, extending possible degenerated zero-volume ells. This 3-step method ahieves aeptable mesh untangling while keeping the mesh lose to the original one. It is suitable for sporadi usage in the ase that omputational mesh degenerates, but not for regular usage for mostly regular meshes. The best strategy in the ALE proess is to keep the mesh smooth during the whole omputation and do not allow suh hardly degenerated meshes. The last approah for mesh smoothing whih is disussed in this study, is the referene Jaobian method introdued in [48], and extended and desribed in more details in [3]. It onstruts referene positions of eah node by loal mesh quality funtional (based on the loal Jaobi determinants minimization. Then, the referene Jaobi matrix depending on the referene nodal positions is onstruted in eah subzone. A global mesh quality funtional based on the Jaobi matries is onstruted and minimized, whih provides the new nodal positions for the new omputational mesh. This final mesh is loser to the original Lagrangian mesh than the referene one, and an be diretly applied in the ALE simulations. 3.3 Conservative Remapping Algorithm The last essential part of the ALE algorithm is the onservative interpolation of all quantities from the Lagrangian omputational mesh, to the new, smoothed one. We require the remapping method to be linearity-preserving (this ondition seems in pratial tests to imply seond order of auray, onservative for all onservative quantities, and loal-bound preserving (the method should not reate new loal extrema in any of the primitive quantities. Our approah results from [68, 7], and redues the problem of remapping all onservative quantities to the problem of remapping of eah of them by a single proess, while satisfying the named properties for all these quantities. In the next subsetions, we desribe the algorithm for remapping of arbitrary funtion (density of a onservative quantity between omputational meshes. In subsetion 3.3.6, we go through the omplete algorithm for remapping of all quantities, where we use the desribed remapping algorithm for eah single onservative quantity. Then, in subsetion 3.3.7, we demonstrate the properties of the remapping method on several numerial examples. Finally, for future use, we extend the idea of the remapping algorithm to 3D and to the ase when the omputational mesh hanges its onnetivity during the smoothing stage. This will allow the future development of the hanging-topology and a 3D ALE odes Remapping of One Conservative Quantity We have two omputational meshes the old one from the last Lagrangian step {}, and the new one oming from the mesh smoothing proess { }. We will denote all quantities related to the new mesh by the tilde aent. Suppose, we have an arbitrary funtion g(z, z = (x, y defined in the omputational domain. The funtion an be arbitrary. In the ontext of ALE methods, this funtion represents the density of mass g = ρ, density of momentum g = ρu, g = ρv, or density of total energy g = ρ(ǫ+ w /. We do not know the expliit formula for the funtion evaluation, we know only the mean values of this funtion in the ells of the original omputational mesh g. These values orrespond to the integrals g = and by the definition of ell mass (or ell momentum or total ell energy, g(x,ydxdy V (3.7 g = G V. (3.73 6

29 We all the integral of the funtion g over the ell as the mass of the funtion g (or ell mass in ell and denote it G (in the ase g(z = ρ(z, it orresponds to the real ell mass G = m. The total mass of the whole omputational domain Ω orresponds to G Ω = g(zdv = g(zdv = G. (3.74 Ω Our task is to ompute the new mean values in the new mesh ells g = G /V. And we want to satisfy the previously mentioned onditions of good remapping method:. Auray we want the new mean values and masses to be as lose to the exat ones, as possible G G exat = g(z dv. (3.75. Conservativity the new total mass must be the same, as the original one G = G Ω. ( Loal-bound preservation we require the new mean values not to reate the new loal extrema, they must remain in the original loal bounds g max g g min, g max = max g, C( gmin = min g (3.77 C( over the neighborhood C( (path of 3 3 ells of the original ell. 4. Linearity preservation in the ase, that the density funtion g orresponds to the global linear funtion, the new mean values and masses must be exat G = G exat = g(zdv, if g(z = a + bx + y. (3.78 Our remapping method is based on the algorithm introdued in [7], [7] and extended in [58]. It onsists of three parts:. Pieewise linear reonstrution inside old ells.. Integration (exat or approximate. 3. Repair. In the first stage, the unknown funtion g is approximated by the pieewise linear funtion, whih is exatly equal to the mean values g in the ell enters (entroids, and linear inside eah old ell. In the seond stage, the reonstruted funtion is integrated over the new ells, whih gives us the new ell masses G, and thus new mean values g. The most natural approah is the exat integration diret integration of the reonstruted funtion over the new ell. Unfortunately, it requires to find all the intersetions of both meshes, whih is omputationally expensive in D. In 3D, this approah is almost unodable and unomputable. Thus, we have developed an approximate integration method swept integration, whih does not require any intersetions, and is easily generalizable to 3D. On the other hand, this method is approximate, and it may happen, that the new mean values violate the loal-bound preservation ondition (3.77. Therefore, we have added the third stage the repair, whih orrets this problem and onservatively returns the values bak to the loal bounds. Finally, our algorithm satisfies all the onditions required for a high-quality remapping method stated above. Let us go through all the remapping stages. 7

30 3.3. Pieewise Linear Reonstrution Pieewise linear reonstrution is the first stage of the remapping proess. Its task is to ompute the slopes of the unknown funtion g in eah ell. Suppose that the reonstruted funtion in ell with ell enter (x,y has the form g (x,y = g + ( g (x x + x ( g (y y. (3.79 y To respet the linearity-preservation ondition of the remapping algorithm, the reonstrution stage itself must be linearity-preserving. There are several methods for funtion reonstrution. We disuss here three of them, the average differene method, the least-squares minimization proess, and finally the Barth-Jespersen limiting proess to enfore monotoniity preservation of the reonstruted funtion. Average Derivative in Neighborhood The most natural approah for the unlimited slopes approximation, is averaging of the funtion derivative over the region V neigh i,j defined by onneting the enters of the adjaent neighbors. The region is shown in Figure 3.4. i,j+ V neigh i,j i,j+ i+,j+ i,j i,j i+,j i,j i,j i+,j Figure 3.4: Path of 3 3 ells and the region of integration V neigh i,j (in dashed lines for the approximation of the average derivative in ell i,j. The unlimited slope approximation in ell = i,j an be written in the form ( g unlim = x i,j V neigh i,j V neigh i,j g x dxdy dxdy. (3.8 The denominator is equal to the neighborhood volume, and the numerator an be redued by using the Green theorem to the boundary integral ( g unlim V neigh g dy i,j = x i,j V neigh. (3.8 i,j 8

31 After defining the funtion value along eah edge of the neighborhood by the average of the orresponding ell values, one an rewrite the whole formula to the form ( g unlim = x i,j = V neigh i,j V neigh i,j e V neigh i,j e V neigh i,j ( g(e, + g (e,+ dy e ( g(e, + g (e,+ ( y(e,+ y (e,, (3.8 where the orientation of the neighborhood boundary edges is assumed in the ounter-lok wise order. This formula an be expliitly expressed for the situation shown in Figure 3.4 as ( g unlim = x i,j V neigh i,j ( (gi,j ( ( ( + g i,j yi,j y i,j + gi,j + g i+,j yi+,j y i,j + ( ( ( ( gi+,j + g i+,j yi+,j y i+,j + gi+,j + g i+,j+ yi+,j+ y i+,j + ( gi+,j+ + g i,j+ ( yi,j+ y i+,j+ + ( gi,j+ + g i,j+ ( yi,j+ y i,j+ + ( gi,j+ + g i,j ( yi,j y i,j+ + ( gi,j + g i,j ( yi,j y i,j, (3.83 where the symbols y i±,j± in this ontext denote the y oordinates of the enters of the orresponding ells. Here, the neighborhood volume V neigh i,j is omputed as the sum of four quadrilateral regions defined by onneting the neighboring ell enters with the enter of the entral ell i,j. The formula for the y slope ( g/ y unlim i,j is derived analogially and has the same form, just a minus sign oming from the Green theorem appears in front of it. This method is straightforward and provides reasonable approximation for the funtion slopes in eah ell. For more details about this approah, see [84], [7]. This method an be used in D, logially retangular meshes. Unfortunately, this method is hardly generalizable to 3D, and several ompliations appear also for the ase of general polyhedral ells. Thus, for future generalization of the ode, we present here also another method based on minimization of the slope error funtional, whih is generalizable to the mentioned, more ompliated ases. Least-squares Minimization Proess The seond method is the least-squares method minimization of the error funtional ( ( g unlim ( g unlim F, = ( g unlim g (x,ydxdy (3.84 x y C( defining the measure of the differene of the neighboring mean values from the (unlimited reonstruted values in the enters of the neighboring ells. Now, let us find the minimum of this funtional (whih means to set the reonstruted funtion as lose to the exat funtion as possible. Let us differentiate this funtional aording to its parameters and set it to zero. We present here the proess of differentiating the funtional aording to the first parameter, V 9

32 ( g/ x unlim, the seond one is analogial. So we solve the following equation ( F = = C( ( ( g x C( (, g y ( unlim g x unlim g unlim g + = C( ( unlim g x (x x + ( R g V g unlim (x,y dx dy V ( unlim g x ( unlim g y (y y dxdy ( (x x dxdy V (3.85 (3.86 and by substituting the definitions of ell volumes (3.3 and ell enters (3.4, (3.5, we get the following expression = ( ( g unlim ( g unlim g g (x x (y y ( (x x. (3.87 x y This equation an be rewritten into the linear form ( g unlim ( g unlim = b x + a xx + a xy. (3.88 x y After repeating the same proess for the y derivative, one an write the omplete system as or expliitly ( axx a xy a xy for unknown partial derivatives, where a yy A a αβ = C( b α = C( ( g z = b (3.89 ( ( g unlim ( x g unlim y = ( bx b y (3.9 (α α (β β (3.9a (α α (g g (3.9b for α,β {x,y}. Solution of this linear system (3.9 is omputed by diret method based on the inverse matrix alulation. The final solution is then ( ( g unlim ( x g unlim = ( bx a yy b y a xy, (3.9 D b y a xx b y a xy y where the determinant of the matrix D = det(a = a xx a yy a xy (3.93 also serves as an indiator of regularity of ell. In some severe fluid motion types it may happen, that the partiular ell degenerates to a line or a point (not exatly, but approximately. This auses the determinant D to be lose to zero and this situation must be treated speially. 3

33 Barth-Jespersen Limiting Proess Finally, to minimize the number of overshoots in the remapped values, we limit the reonstruted funtion. We have tested several most ommonly used limiters [5], and the Barth-Jespersen (BJ approah [8, 7] seems to us to be the most suitable for the reonstrution task. This limiter is onstruted suh that it preserves linear funtion, but it does not allow any overshoots in the reonstruted funtion. The ellular BJ limiter is defined as Φ = min n N( Φn, (3.94 where the nodal limiters are defined as ( g min, max g g Φ n unlim (n g for g unlim (n g > = ( g min, min g g unlim (n g for g unlim (n g < for g unlim (n g =. (3.95 Here g unlim (n is the value of the reonstruted funtion (3.79 omputed with the unlimited slopes ( g/ x,y unlim and evaluated in the nodal position n. The loal extrema g min, g max are omputed as the extrema of the mean values over the neighboring ells (3 3 path of ells around the ell of the partiular ell. The omputation of the unlimited slopes has been presented in the previous paragraphs. The final slopes used for the numerial integration are then omputed as ( ( g g unlim = Φ, x x ( ( g g unlim = Φ. (3.96 y y These slopes guarantee the global linearity and monotoniity preservation of the reonstruted funtion aording to the original mean values g. Formula (3.9 gives us the unlimited slopes, whih is together with the limitation proess (3.96 used for the final slopes evaluation. The main advantage of this approah is the fat that this method is generalizable to 3D meshes, and to meshes with the hanging onnetivity. As for the numerial errors of the final solution, there is no big differene between the unlimited method minimizing the error funtional, and the previous unlimited method of average derivative in the neighborhood, resulting in formula ( Exat Numerial Integration The first approah for getting the new density funtion masses and their mean values, is the exat integration of the reonstruted funtion over new ells. This method analytially integrates the reonstruted funtion over all overlapping elements of both meshes, whih gives us the masses of these ells intersetions. By summing the masses orresponding to some partiular new ell, we get the new ell mass, and after dividing by the new volume, the new density mean value. Let us go through the method in more details. At first suppose, that we have some reonstrution g (x,y as desribed in (3.79 in eah ell, obtained by some reonstrution method desribed in the previous setion. Now, we go through the path of the original mesh fully overing the new ell. Usually, it is pratial to suppose, that the mesh movement was not to severe during the mesh smoothing stage, and one an use the original ell and its nearest neighbors C(. The situation is shown in Figure 3.5. We go through all ells in this path C(, and ompute the intersetion of the ell with the new ell. The examples of suh intersetion of the new ell with the neighborhood of the orresponding original ell, are shown in different shades of gray. Eah intersetion is a polygon, whih an have arbitrary number of verties from (if the ell does not interset the new one, for example ell = i,j + in Figure 3.5 up to 8 (this may happen, when the original ell rotates a little. For omputation of this intersetion region, a robust and preise intersetion method is required. We use an intersetion method desribed in [78], whih is based on the 3

34 i,j+ i,j+ i+,j+ i,j ~ i,j i,j i+,j i,j i,j i+,j Figure 3.5: Original ell = i,j (solid lines and its neighborhood C(, and the orresponding new ell = ĩ, j (dashed lines. All segments of intersetion of the new ell with the original mesh are shown in different shades of gray. omputation of the intersetions of all orretly oriented halfplanes assigned to all edges of both (old and new ells. Some details about the exat integration approah are inluded in [7]. Suppose, we have omputed an intersetion of the old ell and the new ell. We denote this polygon by the symbol P. To ompute the mass in this intersetion region, we integrate the reonstruted funtion of the original ell g (x,y over this region ( ( g G P = g (x,ydxdy = g + x P whih an be rewritten in the form ( ( g G P = g dxdy + P x ( ( g + y P (x x + P P xdxdy x y dxdy y ( g (y y dx dy, (3.97 y P P dxdy dxdy. (3.98 Here, the integrals of, x, and y over the polygon an be evaluated analogially, as in ell volume and ell enters definitions, as we did in setion 3... Using the Green theorem, they an expliitly be expressed from the intersetion region verties positions. Let us show the proess on the example of the volume omputation (integral of. The integral an be redued to the boundary integral, whih an be written as a sum of edge integrals dxdy = xdy = xdy. (3.99 P P e P Now, suppose, that the partiular edge e = [e,e ] with verties e = (x,y and e = (x,y has the following equation x = x + x x (y y, (3. y y whih an be substituted into the integral and evaluated e xdy = x (y y + x x y y y y 3 e x x y y y (y y (3.

35 or after some algebra e xdy = (x + x (y y. (3. The situation of y = y is treated by analogial proess with exhanged x and y oordinates in the edge equation (3.. The final formula is exatly the same as formula (3., in whih is the problemati y y denominator missing, and an be used for arbitrary edge e. The other integrals are evaluated exatly the same way, and an be written as P P P dxdy = xdxdy = P P y dxdy = P xdy = e P x dy = e P (x + x (y y (3.3 6 y dx = e P ( x + x x + x (y y (3.4 6 ( y + y y + y (x x. (3.5 This proess gives us the final expliit formula for the mass of the intersetion region G P by summing them, we get the final formula for new ell mass and new ell density G = C( (3.98, and G P (3.6 g = G V. (3.7 By summing the intersetions volumes (3.3, one an hek, whether some part of the new ell was not forgotten to be inluded (the sum of the intersetion volumes must in some positive ε range orrespond to the new volume V. If we forget some piee, one an extend the intersetion path (level, 3,...neighborhood, ompute the intersetion of the new ell with the larger path, until we over the omplete ell. This situation may our, if the ell hanges signifiantly during the smoothing proess. This method is appliable to our remapping problem and is onservative G = G P = ( G P C( = ( (3.8 G P = G P = G. C( Unfortunately, this method is omputationally expensive due to the need to ompute ells intersetions. Moreover, this algorithm for ells intersetions appears to have problems in ases, that the original and the new ell are almost the same, and their nodes differ just a little (whih happens very often during smoothing in already smooth regions. This algorithm for ells intersetions works only for onvex ells. Another disadvantage of this method is its ompliated generalization to 3D, where planar faes are not guaranteed. Although the method is appliable in 3D [38], the method is unfeasibly slow. Thus, we are fored to develop more sophistiated method, whih does not require the intersetions omputation Approximate Swept Numerial Integration This approah was developed to replae the exat integration method desribed in the previous setion. This method does not need the omputation of the intersetions of the original and the smoothed meshes, 33

36 whih makes this method muh more effiient. This method was introdued in [7] and extended for full usage in [58]. Let us go through this method in more details. Again, we suppose that we know some reonstrution g (x,y (3.79 in eah old ell, oming from some reonstrution method desribed in setion The swept-region integration method is based on the integration of the reonstruted funtion over swept regions, defined by the smooth movement of the ell edges from their old to the new positions. One ell with its swept regions is shown in Figure 3.6. Eah quadrilateral ell has four swept regions orresponding to their four edges. Eah swept region i,j+ i,j+ i+,j+ δ i,j+/ i,j δ i /,j ~ i,j i,j i+,j δ i+/,j δ i,j / i,j i,j i+,j Figure 3.6: Original ell = i,j (solid lines and its neighborhood C(, and the orresponding new ell = ĩ,j (dashed lines. Four swept regions are shown in different shades of gray. is a quadrilateral region surrounded by the original edge, the orresponding new edge, and the lines onneting the orresponding old and new verties. The swept-region algorithm is based on the simple idea, that the mass of the new ell an be omposed from the original ell mass and masses of all four swept regions G = G + δg e, (3.9 or expliitly for the ells enumerated as in Figure 3.6, e E( Gi,j f = G i,j + δg i,j + δg i+,j + δg i,j + δg i,j+. (3. Analogial equation an be written for the new ell volume. When using swept-region masses or sweptregion volumes (or shortly swept masses, swept volumes, we always mean them in signed sense. By definition, if we are in a partiular ell, the orresponding swept regions have positive volumes, if they move outwards from the ell, and negative, if moving inwards. The swept volume V δe is omputed using the same formula as (3.3, from the verties of the swept region. From the sign of eah swept volume we detet, whether the orresponding edge moves inwards or outwards of the original ell. This approah naturally handles twisted swept regions (as the upper swept region δ i,j+/ in Figure 3.6, whih may our in ase, that one edge vertex moves inwards and seond one outwards of the ell. In this ase, the sign of the swept volume orresponds to its larger segment, whih overweights the smaller one. For omputation of the swept mass, we integrate reonstrution from the ell, in whih more of the region lies { for Vδe = (3. for V δe >, 34

37 where the ell here is a neighbor of ell, whih have a ommon edge e with. The integration is straightforward, δg e = g (x,ydxdy δ ( e ( ( g g = g x y dxdy (3. x y δ ( ( e g g + xdxdy + y dxdy, x δ e y δ e where the integrals are omputed the same way as shown in equations (3.3, (3.4, (3.5. Finally, we add this swept mass to the mass of the partiular ell, and subtrat it from the orresponding neighbor. Due to this edge-based nature of the algorithm, eah swept volume and swept mass an be omputed only one, and used twie for both adjaent ells, whih inreases its effiieny. Moreover, due to this fat, the algorithm is naturally onservative the ell masses are interhanged between neighboring ells, what is added in one ell, is subtrated from other one. As for linearity preservation, this algorithm exatly integrates the linear reonstruted funtion (in ase of global linear funtion, the reonstrution is also a global linear funtion, so the produed new masses (and thus new densities are also exat. So the swept integration method preserves global linear funtion as well. This property, together with the smooth underlying funtion g, implies the seond order of auray [7]. The swept region integration is an approximate method. In the regions of signifiant hanges of the remapped quantities (typially, the regions lose to shok waves, it may happen, that the new density value overjumps the loal extrema of the original mean values. Thus, one more step must follow, to enfore the loal-bound preservation ondition Repair Repair is the last stage of the remapping proess, whih orrets the possible loal-bound violations aused by the integration proess. This method an be desribed as the onservative redistribution of the onservative quantity. It was introdued in [58] and extended in [87], where its appliation to advetion equation is shown. The first step of the repair proess should be finding the bound-determining neighborhood of ell, the piee of the original mesh fully overing the new ell, and omputing loal extrema of the underlying funtion from the original mean values in this bound-determining neighborhood. Finding the real bound-determining neighborhood would require omputations of the intersetions of the original and new mesh, whih we want to avoid. In pratial implementation, we use the same loal extremes, whih we omputed during reonstrution limiting proess in setion These extrema are omputed over the ell neighborhood C( (3 3 path of ells, whih overs the whole bound-determining neighborhood for reasonably strong mesh movement. So, in eah ell, we know the original mean value g, the new mean value g, and the loal extremes g min g max If the new mean value lies within the loal extremes g min = min C( g (3.3 = max C( g. (3.4 g g max, (3.5 the ondition is satisfied and no repair proess is required. Suppose, that the loal minimum is violated g g min, (3.6 35

38 maximum ase is treated analogially. At first, we ompute the mass, whih is missing in the new ell to bring the value bak to the loal minimum δg need = ( g min g V (. (3.7 The algorithm has to be onservative, so we annot just add the mass to the ell, we have to subtrat it from the mass of the neighborhood. For eah neighboring ell, we ompute the mass δg avail = max (,(g g min V (, (3.8 whih an safely be taken from it without violating its loal minimum. Then, the total available mass in the neighborhood is their sum δg avail C( = δg avail. (3.9 C( If the available mass is smaller than the needed mass δg avail C( < δgneed, (3. we have to extend the neighborhood stenil and searh for the mass in a larger region. If we found enough mass δg avail C( δgneed, (3. we an perform the repair proess. We inrease the wrong new value up to the loal minimum G = g min V (, (3. and take the mass, whih was just added to it, from the neighboring ells proportionally to their available masses G = G δgavail δg avail δg need (3.3 C( from all ells from the neighborhood C(. This method repairs all loal-bound violations oming from the integration proess. In [58], we have also proved, that this proess suessfully finishes the repair in a finite number of steps. It is not obvious, that the repair algorithm always finds enough mass in some (maybe larger neighborhood, so let us go through the proof to lear it here. Suppose, that the new ell is fully overed by the neighborhood C( of the original ell, if and only if C(, (3.4 and suppose, that the loal minimum is violated in ell (as in equation (3.6. Let us define two quantities, M = ( g min g V ( (3.5 M + = :g <g min :g >g min ( g g min V (. (3.6 Here, M > is the total amount of mass, that is needed to bring all new densities in ells, where they are lower that their loal minima g min, up to these loal minima. On the other hand, M + > is the total amount of mass, that an safely be taken from all other ells without violating their loal minima. To show, that the repair proess is suessful, we have to prove the following relation M M +. (3.7 36

39 Let us write the total mass in the entire mesh as M = = = g V g min V ( + :g >g min g min V ( + M + M, ( g g min V ( :g <g min ( g min g V ( (3.8 whih means the total mass of the loal minimal masses, plus the total masses over this minimum, minus the loal masses missing in these loal minima. One an rewrite the formula as M + M = M g min V ( (3.9 and using the definition of g min M + M = M = M and assumption (3.4, one an rewrite it as g min V ( M g V ( C( C( g V ( = M V ( = M g V ( = M M =. g (3.3 That is, M M +, and thus we proved, that it is always possible to finish the repair proess by extending the repair stenil. The presented repair proess is naturally onservative we are just exhanging the mass from one ell to another, no new mass is added or taken away. Moreover, this algorithm does not affet the linearity preservation of the omplete remapping proess. In the ase of global linear funtion, the loal extrema are situated in the neighboring ells, not in the atual one, and the repair proess is not turned on at all. Problems may only appear in the regions lose to the domain boundary, where the loal extremes must be treated by suitable boundary onditions by extrapolating the global underlying funtion to the external ghost ells Remapping of All Quantities In this setion we disuss the algorithm for remapping of all onservative quantities from the Lagrangian omputational mesh, to the new, smoothed one. The omplete algorithm is based on an older version of the remapping algorithm presented in [7]. The following properties are required from the algorithm:. Conservation the total mass, momentum in both diretions, and total energy must remain unhanged. This together with the Lagrangian step onservativity ensures the onservation of the omplete ALE algorithm.. Loal-bound preservation the remapped mass density, veloities in both diretions, and internal energy, must remain in the loal bounds from the original values. This is important for redution of possible generation of new peaks and osillations during the remapping stage. 3. Auray by auray, we mean in this ontext, that the remapping proess preserves any linear funtion. This property orresponds in pratial tests to the seond order of auray. In [7], more auray issues are disussed. 37

40 4. Reversibility this property says, that in the ase of idential Lagrangian and new mesh, the remapped values must remain unhanged. This natural and important property is related to the ontinuous dependene of the hange of primary quantities between the old and the new mesh [7]. We disuss here suh remapping algorithm and desribe all the important onerns of it. We use the remapping algorithm for onservative remapping of density of one arbitrary onservative quantity inside this setion, whih is desribed in previous setions. Let us remark, that satisfation of the mentioned required features is mostly implied by their satisfation by the method for remapping of one quantity. The omplete algorithm for remapping of all quantities to the new mesh an be desribed in the following way. At first, we define all the needed quantities in the mesh subzones. This an be understood as defining the quantities on a double-dense mesh (mesh of subzones, four values in eah ell. This stage is alled a gathering stage in [7]. It is important do define the subzonal quantities onservatively, suh that their total value in the zones is equal to their sum over the orresponding subzones. The masses m n (and thus densities ρ n are already defined in subzones, we need to define also subzonal momentum densities in both diretions µ n, νn (and thus subzonal veloities un, vn, and the subzonal internal and kineti energy densities E n, K n. Definitions of all mentioned subzonal quantities orresponding to the subzone of ell at node n are presented as follows ρ n = mn V n (3.3 µ n = ρn u n (3.3 ν n = ρn v n (3.33 E n = ρ n ǫ (3.34 K n = ρn (u n + v n. (3.35 The definitions of subzonal veloities and speifi internal and kineti energies are obvious w n = w n (3.36 E n = ǫ (3.37 K n = (u n + v n. (3.38 It is easy to hek, that this distribution of the onservative quantities to subzones is onservative. Now omes the remapping stage, in whih we remap eah subzonal onservative quantity from the Lagrangian mesh to the smoothed one. The first omes the subzonal density of mass ρ n, whih is remapped to the new values ρñ, and then repaired (for explanation see setion aording to the loal extrema omputed from subzonal densities in subzones neighboring the atual one. Now, one an update the new zonal, nodal, and subzonal masses, and zonal densities mñ = ρñ V ñ (3.39 m = mñ (3.4 ñ N( mñ = C(ñ mñ (3.4 ρ = m V. (3.4 Now, let us have a look at the momentum remapping. Using the same proedure for one onservative remapping, we remap both momentum densities µ n, νn to the new mesh and get new momentum µñ, 38

41 νñ. This allows us to define new subzonal veloities uñ = µñ ρñ vñ = νñ ρñ (3.43 (3.44 and the new nodal veloities wñ = mñ C(ñ ( m ñ wñ. (3.45 Now, from the new nodal densities ρñ = mñ/vñ and the original nodal veloity loal extremes, one an ompute the nodal extremes of the momentum density µ min ñ = ρñ u min n, (3.46 and analogially the others. Using these extremes, one an repair the nodal momentum densities and then reompute the new veloities bak µñ = ρñ uñ (3.47 νñ = ρñ vñ (3.48 uñ = µ ñ ρñ (3.49 vñ = ν ñ. (3.5 ρñ This auses the new nodal veloities to satisfy the loal-bound preservation ondition. The last quantity we need to remap, is the density of the subzonal total energy T n = En + Kn, (3.5 whih gives us the new subzonal total energy T ñ. From the new density and new veloities, we ompute the new density of kineti energy Kñ = ( ρñ u ñ + vñ (3.5 and the new subzonal and zonal density of internal energy Eñ = T ñ Kñ (3.53 E = Eñ V ñ. (3.54 V ñ N( From the original ellular speifi internal energy ǫ, we ompute loal extremes of the speifi internal energy ǫ min and ǫ max, and transform them to extremes of density of internal energy E min E max = ρ ǫ min (3.55 = ρ ǫ max. (3.56 Then, we perform the repair of densities of the ell internal energy E using to these loal extremes. This proedure auses the final speifi internal energy ǫ = E ρ (

42 g g to satisfy the loal-bound preservation ondition. Now, we an update the ell pressure field from the equation of state, apply the pressure boundary ondition, and we have the omplete algorithm for the remapping of all onservative quantities from the original Lagrangian mesh to the new, smoothed one. If the remapping method for one quantity satisfies all the required properties (whih our method desribed in the previous setions does, this omplete algorithm satisfies them also. Compared with the method desribed in [7], there are several differenes in our approah. In fat, we are using an older version of the method [7], whih stritly separates three stages of the algorithm gathering stage defining all subzonal quantities, remapping stage performing one by one subzonal quantity remapping and repair (all quantities are repaired in subzones, and the sattering stage defining the final nodal and zonal quantities. Next differene is in the fat, that the algorithm in [7] remaps the kineti and internal energies separately, not their sum the total energy. The separate energy remapping should be better, the algorithm does not mix two quantities, whih may vary in an order of magnitude. We plan to add this property to the remapping algorithm in the future version. The last important differene ours in the omputation of the subzonal veloities. In our algorithm, there is no prinipal differene between nodal and subzonal veloities, whih is orret in sense of onservativity, but it may derease the auray of veloity remapping. In [7], there exist a transformation matrix in eah ell, whih onverts the nodal veloities to subzonal ones, and vie versa. This requires omputation of the inverse matries, involving lot of additional omputations. Moreover, this approah an generate onsequent osillations in the veloity field, and more repair steps would be neessary for orreting this problem Numerial Testing of Remapping Methods In this setion, we present several numerial examples showing properties of our omplete remapping algorithm. Thus, we demonstrate all important features of the remapping algorithm onservativity, linearity preservation and loal-bound preservation. We fous on the importane of the limiting proess in the reonstrution stage, and the repair proess. The first numerial example, we demonstrate here, is a linear funtion g(x,y = x + y (3.58 remapped over a series of randomly hanging quadrilateral meshes of 6 ells in, omputational domain. In Figure 3.7, one an see the remapped values on the final mesh (a, the front view (b, and Y y x x X (a (b ( Figure 3.7: Linear funtion remapped over a series of randomly hanging omputational meshes. (a view to the final mean values, (b front view, ( ontour map. their ontours (. The desribed method, using Barth-Jespersen limited reonstrution, swept region 4

43 g g integration, and the final repair proess, is used. This example demonstrates preservation of the global linear funtion, numerial error is zero up to the round off error. In the seond example, we use the following sine funtion g(x,y = + sin(π x + sin(π y (3.59 remapped over a series of 8 orthogonal, sine-like moving meshes desribed in [58]. The position of node n = i,j in time t of the omputation is desribed by the following formulas x i,j (t = ( α(tξ i + α(tξ 3 i (3.6 y i,j (t = ( α(tη j + α(tηj (3.6 α(t = ( sin 4π t, (3.6 where the logial oordinates ξ i = i/n x, and η j = j/n y. Clearly, for t = and t = t max, the parameter α(t =, and the initial and final meshes are idential and uniformly retangular. In Figure 3.8, the t max Y y x x X (a (b ( Figure 3.8: Sine funtion remapped over a series of orthogonal, sine-like moving omputational meshes. (a view to the final mean values, (b front view, ( ontour map. mean values (a on the last omputational mesh of 6 ells in, domain, the front view (b, and the ontour maps ( are shown, omputed using the full limiting+swept integration+repair method. The initial and the final meshes are the same, whih allows us ompare the numerial errors of the remapping. In Table 3., relative numerial errors L and L max of the desribed problem, depending on the number L N x = N y N t L Nx L Nx L max L Nx max L Nx max e e e- 6..e e e e e e-5 5.7e-3 Table 3.: Numerial errors of the remapping proess inluding limited reonstrution, swept region integration, and the repair stage applied to the smooth sine funtion (3.59. L and L max errors for different mesh sizes and different number of remappings, and their ratio to the errors with half number of mesh ells in eah diretion, are presented. of omputational ells N x N y, and the number of remaps performed N t, are presented. The L error 4

44 shows at least seond order of auray of the omplete remapping method, the maximum L max error slowly onverges to the seond order of auray too. The last numerial test presented here is remapping of a non-smooth square funtion { ( for (x g(x,y =.3 ( (y.3 (3.63 else over a series of onsequently smoothing meshes. This funtion inludes a strong jump, and is hard for the repair algorithm to be treated in a good quality. The original mesh is a randomly distorted mesh, whih is smoothed into the next step by plain averaging (3.66. The final mesh is obtained after smoothing steps. In Figure 3.9, we show the mean values in the final, almost smooth mesh, using different ombinations of limiting and repair methods with the approximate swept region integration. Relative numerial errors and the global extrema of all the methods on mesh with 6 6 ells, are presented in Table 3.. From the presented numbers, one an see that all the numerial errors are reonstrution and repair L L max g min g max T[s] swept, unlimited, no repair 9.97e-3.5e- -5.9e swept, unlimited, repair 9.85e-3.9e-.4 swept, BJ limiter, no repair.6e-.93e e swept, BJ limiter, repair.6e-.93e-.8 exat, BJ limiter, no repair.3e-.98e-.784 Table 3.: Numerial errors L and L max and the global extrema g min and g max of the different remapping methods applied to the non-smooth square funtion (3.63 in 6 6 ells mesh after steps. Time of omputation T in seonds on a.4ghz mahine is also presented. omparable, and in this partiular ase, the swept integration method is even better than the exat one. The limiting proess dramatially dereases number of possible overshoots, but does not guarantee absolute bound preservation. The repair proess helps in every situation, but due to the higher number of repairs, it auses the whole omputation to need more time. From the times of omputation, one an see that all swept integration simulations are muh faster than the exat one. Thus, all parts of the remapping method are essential. The swept integration is muh more effiient than the exat one, the limiting proess dereases the number of repairs applied, and the repair stage must be performed to guarantee bound preservation. Only ombination of all these three stages guarantees optimal results for non-smooth underlying funtion. For more numerial tests of the D remapping algorithm, see [58] Remapping in 3D For the future development of the 3D ALE ode, we have developed a 3D version of the omplete remapping algorithm for arbitrarily polyhedral ells. The 3D algorithm was desribed and several preliminary examples presented in [35], and the full algorithm desription with omprehensive numerial tests is presented in [36]. Here, we briefly desribe the main differenes of the 3D remapping proess from the D ase. The main problem, whih we have to deal with in 3D, is the fat that the faes in 3D do not have to be planar. Only the positions of the fae nodes are known, the faes themselves (their shapes are not defined. For eah fae f of a partiular ell, we find the fae enter by plain averaging all fae verties. By onneting all fae verties to this fae enter, we uniquely subdivide the fae to triangles orresponding to eah fae edge. This way, every faes shape is uniquely defined. In 3D, we use exatly the same proess of error funtional minimization for underlying funtion reonstrution, as desribed in setion In 3D, although the exat integration method was developed 4

45 g g g g g g g g Y (a y x x X Y (b y x x X Y ( y x x X Y (d y x x X Figure 3.9: Non-smooth square funtion (3.63 with its front views and ontour maps, remapped over a series of onsequently smoothing meshes to the final mesh. Several ombinations of reonstrution and repair, with the swept integration, are used (a unlimited slopes, (b unlimited slopes + repair, ( BJ limiter, and (d BJ limiter + repair. 43

46 [38], it is quite a ompliated and unfeasibly omputationally expensive proedure. Thus, for pratial use, we have only the swept integration method available. As for the swept integration and the repair algorithm, there is no major priniple differene from the D ase. The only really diffiult part, is the integration of the linear funtion over arbitrary polyhedron, whih is needed for the omputation of the ell volumes V ( = dxdy dz, (3.64 ell enters x = V ( xdxdy dz, y = V ( y dxdy dz, z = V ( z dxdy dz, (3.65 swept volumes, and masses analogial to D swept masses (3.. This integration proedure is onsuming most of the omputational time. Let us desribe the integration proess briefly. For the integration of the linear funtion over arbitrary polyhedron with faes split into triangles as desribed before (for example ell, we use the method introdued in [74]. Let us demonstrate the integration proedure on the example of the omputation of the ell volume, whih an be written in the form V ( = dv = divφ dv = Φ n + ds, (3.66 where the vetor funtion Φ have the form Φ = 3 (x,y,zt, (3.67 and n + is the outer normal of the ell. The surfae integral an be represented as sum of the fae integrals, whih an be omposed as the sums of integrals over orresponding triangles. Therefore, the previous expression for volume an be rewritten in the following way V ( = n +,κ ( κds, ( f F( T(f κ=x,y,z where n +,x, n +,y, n +,z, are omponents of the normal n + ( orresponding to the triangle, whose orientation is onsistent with outward normal to boundary of ell. The symbol F( denotes the set of all faes of ell, and T(f denotes the set of all triangles of fae f. To evaluate the integrals κds (3.69 for κ = x,y,z, we projet eah triangle to one of the oordinate planes, as desribed in [74]. Suppose, that the triangle belongs to the plane n +,x ( x + n +,y ( y + n +,z ( z + ω =, (3.7 where ω = n + p and p is an arbitrary point in the plane (for example, one of the verties of the triangle. To redue the error of omputations, for the given triangle we hoose α β γ as righthanded permutation of x y z oordinates, suh that n +,γ is maximal. If we denote projetion of the triangle to (α, β plane by Π, then integrals over an be redued to integrals over Π as follows α ds = n +,γ J Π α (3.7a β ds = n +,γ J Π β (3.7b γ ds = n +,γ (n +,γ (n +,α J Π α + n +,β J Π β + ω J Π, (3.7 44

47 (a (b Figure 3.: Distribution of the linear funtion (3.74 in the original twisted mesh (a, and the final RJ method smoothed mesh (b. In both pitures, ut in z = plane is shown. where the integral J Π g is defined as J Π g = g dα dβ. (3.7 Π Finally, using the Green theorem in (α, β plane, the integrals Π g dα dβ of polynomial funtion g are redued to the D integrals over the edges and evaluated in the form J Π α J Π β = 6 sign(n+,γ e = 6 sign(n+,γ e J Π = sign(n+,γ e δβ e (α e + α e α e + α e δα e (β e + β e β e + β e (3.73a (3.73b (δβ e (α e + α e δα e (β e + β e, (3.73 where eah edge e has the end points [e,e ] in α,β oordinates, and δα e = α e α e and similarly δβ e. The final formula is then the ombination of (3.73, (3.7, and (3.68. Computation of integrals of arbitrary linear funtions follows the same path, just the vetor funtion Φ has to be redefined. For example, in the ase of the omputation of the integral of x oordinate (x from equation (3.65, we use the form of Φ x = ( x,, T suggested in [74]. Then, in the final stage one needs to ompute D integrals of higher degree polynomial funtion, whih still an be done expliitly. For ompleteness, we present here one 3D numerial example of the remapping proess. To show the linearity-preservation property of the algorithm, we have hosen the example of a linear underlying funtion g(x,y,z = + 3x + y + z (3.74 remapped over a series of 3D polyhedral meshes inluding about 7 different polyhedrons with up to faes in a unit sphere. Using the Referene Jaobian mesh smoothing method (RJM desribed in setion 3.., we get a series of 9 onseutive meshes the original twisted one, and 8 gradually smoothing meshes. Eah mesh is obtained from the previous one by applying steps of the referene Jaobian smoothing proess. In Figure 3., we present the ut in z = plane through the original twisted unit-sphere mesh, and the final RJM smoothed mesh. The remapped mean values of the linear 45

48 (a (b ( Figure 3.: One reonnetion in D mesh (a. Solving in two steps: shrinking old edges to one point (b and expanding it to the new edges (. Original mesh in dashed lines, new one in solid lines. Swept regions are shaded in gray. underlying funtion (3.74 orrespond exatly to the values of the funtion in the new ell enters, the numerial error is (up to round-off error zero. For more numerial examples of 3D remapping, see [36] Remapping with Changing Topology Next useful generalization of the remapping method for the future, is remapping from the Lagrangian mesh to the rezoned (smoothed mesh with different topology. This is an important issue, beause some of the mesh rezoning methods are not based on a pure nodal movement, they an also hange the onnetions between them. We have developed generalization of the desribed remapping algorithm to the ase of meshes with hanging topology [57], whih satisfies the same properties (onservativity, reversibility, linearity and loal-bound preservation. Here, we only summarize the main ideas of this method. In topology hanging ase in D, there is no differene in reonstrution and repair stages, there is no need to hange anything. We have the exat integration method available without any hange, there is only a problem with seleting the path of the original mesh for intersetion omputation. The only requirement is, that both meshes must have onsistent ells enumeration orresponding ells in both meshes must have the same index to identify the mirror of eah ell in the rezoned mesh. The same problem is with the mesh path for reonstrution limiting, and for repair stage. Determining of suh ell neighborhood (alled good neighborhood is quite a ompliated task [9]. For this partiular ase, we swithed from lassial neighborhood identifiation (just neighbors of ell to geometrial neighborhood identifiation. In this approah, we find the path as ells, whih are geometrially (not topologially lose to the original one. The swept region integration must be revised for hanging topology. The typial situation, when one reonnetion appears, is shown in Figure 3.. Detetion of suh reonnetion is easy we have to detet two ells, whih were neighbors in the original mesh, and are not neighbors any more in the rezoned mesh. In the example shown in Figure 3. (a, these are the left and the right ell. We detet the edge, whih is removed from the original mesh, and the edge, whih is new in the new mesh. We perform the swept integration in two steps. In the first step, we shrink the disappearing edge into one entral point omputed as an average of the nodal positions of verties of both edges as shown in Figure 3. (b. We have to perform five swept integrations orresponding to the disappearing edge, and four edges 46

49 (a (b ( Figure 3.: Linear funtion on the initial quadrilateral mesh (a, remapped to the entral polygonal mesh (b, and remapped to the final mesh. onneted to it. In the seond step, we extend the entral point into the new edge doing similar next five swept integrations shown in Figure 3. (. This proess treats the situation of one topology hange in the ell. Speial are must be taken, when the reonnetion appears lose to the domain boundary. If there should appear more reonnetions in one ell, it an be solved by reduing the smoothing, and performing the mesh smoothing tehnique just to the point, when one reonnetion appears, and then ontinue to the next one. We present here one numerial example of remapping with hanging topology. As the example, we demonstrate remapping of a linear funtion over a series of Voronoi meshes. The initial mesh is a quadrilateral mesh of 3 ells, whih is in fat equal to the Voronoi mesh with equidistant generators. We move the generators aording to the sine movement, desribed in [58]. Figure 3. (a, presents the linear funtion distribution on the initial quadrilateral mesh. In the middle of the omputation, after a series of remappings, we get the situation shown in Figure 3. (b, where the mesh has ompletely different onnetivity. The final mesh in Figure 3. ( is the same, as the original one, and the remapped mean values orrespond exatly (up to round off error to linear funtion values in the ell enters. Massive hanges of mesh topology were done during the proess, the middle-time mesh (b is ompletely different, from the initial (final one, and remapping still works well. For more numerial tests, see [57]. 47

50 Chapter 4 ALE Method in Cylindrial Geometry In this hapter, we desribe hanges in the ALE method, whih are needed to perform simulations in the D ylindrial geometry. By the D ylindrial geometry, we understand here, that the simulated problems have ylindrial symmetry, the solution depends on the distane r from the axis z, and the position in z diretion only. There is no dependene on the ylindrial angle. 4. Lagrangian Solver There exist several ylindrial Lagrangian approahes, we desribe two methods from [8] the Area- Weighted Differening (AWD sheme and the Control Volume (CV method. Both these methods have the same staggered disretization, as in the Cartesian ase. Let us desribe both generalizations of the Lagrangian method to the ylindrial oordinates briefly in the following subsetions 4.., 4... At this moment we note, that the AWD sheme is presented here just for ompleteness. Due to its geometrial nature, it is not possible to ombine it into a omplete onservative ALE algorithm. On the other hand, the sheme itself is quite simple and easy to understand, so we do not skip the AWD method here. 4.. Area-Weighted Differening Sheme The omplete derivation of the Area-Weighted Differening (AWD sheme in ylindrial oordinates is presented in details in [8] as an example of numerial method preserving spherial symmetry of the solution. As for the geometry used for the AWD approah, the ell enters r, z are omputed as the arithmetial averages of the orresponding ell verties. Similarly, edge enters are omputed by plain averaging of its verties. Now, we ompute Cartesian volumes of the dual triangles A l orresponding to all edges in eah ell, as shown in Figure 4.. The subzonal volume of the lth subzone of ell is defined as V l = r 5A l + 5A l + A l+ + A l+ n(,l, (4. where r n(,l is the r oordinate of the node n(,l, belonging to the partiular subzone. The zonal volume an be written in the form V = 4 l= V l = 4 l= A l r n(,l + r n(,l+ + r 3, (4. as a sum of dual triangles with the r fator (average of r oordinates of all verties of the partiular triangle. Cellular, nodal, and subzonal masses are omputed from the initial subzonal densities and the desribed volumes, using the same formula as in the Cartesian ase, (3.9, (

51 i,j e 3 i,j 4 A 3 3 e 4 A 4 =i,j A e A i,j e i,j Figure 4.: Cell in AWD approah with dual triangles A l (shaded in different grades of gray orresponding to all ell edges e l. Cell subzones are enumerated from to 4. In the AWD approah, the momentum equation has the disrete form ( w m n = r n F n, (4.3 t and the energy equation has the form ( ǫ m = t 4 l= n r n(,l ( u n(,l F r l + v n(,l F z l, (4.4 where the veloity vetor w = (u, v have its diretion in r, z oordinates, and the vetor of orner and total nodal fores are omputed using exatly the same proess, as desribed in setion 3..6 for Cartesian geometry. When we ompare the equations with the Cartesian equations (3.5 and (3.54, one an see the obvious generalization of the Cartesian method to ylindrial geometry. The orner fores F l = Fp l + Fdp l + Fq l (4.5 are omputed exatly the same way as in the Cartesian geometry, using the same formulas for zonal pressure (3.8, subzonal pressure (3.8, and visosity fores. Then, just by multiplying these fores by the r oordinate of the orresponding node, we onvert these fores to the ylindrial, AWD fores. The AWD algorithm provides reasonable solutions in the ylindrial geometry, whih are onservative, ylindrial symmetry preserving, and onsistent with the fluid equations. Unfortunately, due to the volumes definitions, it is not possible to onsistently ombine this approah with the remapping stage into the omplete ALE algorithm, suh that the algorithm would be onservative. The reason is, that the remapping algorithm requires ell masses (and volumes to be defined in suh a way, that they an be omputed as integrals of some quantity (density, over the ell volume. In the AWD approah, the ell volume is defined as a sum of triangle volumes multiplied by the r fator, whih annot be transformed to any integral formula. Thus, we have to leave the AWD approah, and searh for a different method, where the ell (and subzonal volumes are defined as integrals. 4.. Control Volume Method The seond method, whih an be used for the ylindrial Lagrangian solver, is the Control Volume (CV method shown in [8]. This method has one disadvantage it does not preserve the spherial symmetry 49

52 (for more details see [8], some more details were provided by [67]. On the other hand, this property at the moment is not so important in our real alulations, it is fully overweighted by the fat, that the volume is naturally defined as the integral in ylindrial oordinates over the ell, whih is needed for the ombination with the onservative remapping method in the ALE ode, as desribed in the last paragraph of the previous setion. As for the geometry (defined by the method of volume evaluation in the CV method, the ell volumes are defined as ylindrial integrals of over the ell V = r dr dz, (4.6 whih an be redued by the Green theorem to the edge integrals and evaluated from the oordinates of the edge verties V = r e e dz = 4 (z l+ z l ( rl 6 + r l+ + r l r l+. (4.7 l= Here (r l,z l are oordinates of node n(,l in (r,z oordinate system, where the nodes are enumerated as introdued in Figure 3.. The subzonal volumes V l are omputed exatly the same way, and due to to the volume defined by the analytial integration, learly V = 4 l= V l. (4.8 Analogially, the ell enters are defined as ylindrial integrals of the orresponding oordinate over the ell r = r r dr dz = r 3 dz (4.9 V V e e 3 z = z r dr dz = r z dz, (4. V V whih an be evaluated from the edges verties similarly. In [8], arithmetial averages of ell verties oordinates are used for the ell enters omputation, but we need the entroids (4.9, (4. for the ombination with the remapping stage in the ALE method. Our numerial tests show, that this algorithm hange does not affet the numerial stability nor the order of auray of the ylindrial CV method. Let us derive the ylindrial CV fores, as we did for the Cartesian geometry in setion 3... The momentum equations has the following general form ρ d w = p, (4. dt not depending on the oordinate system. Gradient of arbitrary funtion in the ylindrial oordinates an be written as f = gradf = ( f r, r e e f ϕ, f z, (4. or applied to the pressure p in the ylindrial symmetry (ylindrial symmetry assumes, that the funtion does not depend on ϕ we get ( p gradp = r,, p, (4.3 z where the derivatives an be written as p z = (p r r z p r = ( (p r r r (4.4 p. (4.5 5

53 The formulas are equivalent, the right hand side an be redued to the left hand side form, we need it in this speial form for further derivations. Now, the original system (4. with w = (u,v an be rewritten as ρ du dt = r ρ dv dt = r ( (p r r (p r z p (4.6. (4.7 After multiplying the equations by r and integrating over the nodal region V n (the same as the Cartesian nodal region from Figure 3.3, we get the ylindrial nodal fores on the right hand side of both equations. Let us fous on the fore in the r diretion at first. By assuming a onstant pressure value in eah subzone, it an be written as Fp r n + Fdp r (p r (p r n = dr dz + p dr dz = dr dz + V n r V n V n r 4 l= V l n p dr dz, (4.8 and using the Green formula, one an redue both integrals to the boundary integrals to the form Fp r n + Fdp r n = p r dz + V n 4 l= p n (n,l V l n r dz, (4.9 where the seond integral was deomposed to the orresponding integrals over all subzones adjaent to the node n, and the subzonal pressure was moved in front of the integral. Both boundary integrals an be written as a sum of edge integrals, and the edge pressure is defined as the average of the subzonal pressures in both adjaent subzones (as in the Cartesian ase. The formula is then Fp r n + Fdp r n = 4 ( pl (n,l + pl+ (n,l l= s (n,l l ( 4 + r dz + l= p n (n,l s (n,l l a ± l r dz + pl (n,l + pl (n,l s (n,l l s (n,l l r dz r dz + a (n,l l r dz a (n,l+ l r dz. (4. After substituting the formula (3.4 for the subzonal pressure, and after evaluating the integrals I (s l = r dz = ( r e(,l + r z z e(,l (4. s l I ( a ± l = r dz = ( r e(±,l + r n(,l z e(±,l z n(,l, (4. one an write the fore as Fp r n + Fdpr n ( = 4 (p (n,l + δpl (n,l + δpl+ (n,l + l= 4 l= ( ( p (n,l + δp l (n,l I(s (n,l l I(s (n,l l + (p (n,l + δpl (n,l + δpl (n,l I(s (n,l l + I(s (n,l l I(a(n,l l + I(a (n,l+ l. (4.3 5

54 By rearranging all the terms, we obtain the fore in the form Fp r n + Fdp r n = 4 ( p (n,l l= ( δp (n,l l= 4 l= I(a (n,l+ l I(a (n,l+ l I(a (n,l l I(a (n,l l ( I(s (n,l l δp l+ (n,l δpl (n,l + 4 l= ( I(s (n,l l δp l (n,l δpl (n,l, (4.4 whih orresponds exatly to the Cartesian formula (3.8. Obviously, the first term of the formula orresponds to the zonal pressure fore in the ylindrial oordinates. The nodal fore in z diretion is omputed similarly. From (4.7, one an express the fore in the form Fp z n + Fdp z (p r n = dr dz, (4.5 V n z and after applying the Green formula, one an rewrite it as a boundary integral Fp z n + Fdp z n = p r dr. (4.6 V n The sign hange arises from the minus sign in the Green formula for the seond oordinate. As in the previous ase, we deompose the boundary integral to the sum of edge integrals, and define the value of pressure at the edge to the average of the pressures in the adjaent subzones, and rewrite the formula in the form Fp z n + Fdpz n = 4 l= ( pl (n,l + pl+ (n,l s (n,l l r dr + pl (n,l + pl (n,l s (n,l l r dr. (4.7 Again, we an rewrite the subzonal pressures as variations of the zonal pressures, and use the following symbols J (s l = r dr = ( r e(,l + r r r e(,l (4.8 J ( a ± l = s l a ± l r dr = ( r e(±,l r n(,l r e(±,l + r n(,l for the edge integrals, whih allows us to rewrite the fore to the form (4.9 Fp z n + Fdp z n = 4 ((p (n,l + δpl (n,l + δpl+ ( (n,l J l= s (n,l l (p (n,l + δpl (n,l + δpl ( (n,l J s (n,l l. (4.3 For further hanges, we need the assumption of the losed subzone l of ell (n,l, whih implies the following equation ( ( ( ( ( = r dr = J + J J + J, (4.3 or V l (n,l ( J a (n,l+ l a (n,l l ( J a (n,l l a (n,l+ l ( = J 5 s (n,l l ( J s (n,l l s (n,l l s (n,l l. (4.3

55 This equality allows us to express the final z fore (after some regrouping of terms as 4 ( Fp z n + Fdpz n = + p (n,l + + l= 4 ( δp (n,l l= 4 l= J(a (n,l+ l J(a (n,l+ l J(a (n,l l J(a (n,l l ( J(s (n,l l δp l+ (n,l δpl (n,l + 4 l= ( J(s (n,l l δp l (n,l δpl (n,l, (4.33 whih is the same formula as the one for the r fore (4.4, with Js instead of Is. The ylindrial fores are omputed using exatly the same formulas, as the Cartesian fores (3.8. The only differene is in the evaluation of the I and J integrals. When we ompare the Cartesian (3.7, (3. and ylindrial (4., (4.8 I, J integrals, there is the additional r fator equal to the average of the r oordinates of the edge verties in the ylindrial ase. The rest of the ylindrial Lagrangian step an remain unhanged, the r fator in the ylindrial integrals and volumes auses that the CV method approximates the ylindrial equations. There is still the question of the ylindrial visosity fore unresolved. Our approah, inspired by [67, 85] is based on the evaluation of the visosity fores using exatly the same Cartesian formulas, as desribed in setion Then, we add the r fator to these final fores Fq l = rl Fqart l, (4.34 where the enter of the subzone r l is omputed using formula (4.9 applied to the subzone. This modifiation is suffiient for the fully working CV algorithm in ylindrial oordinates. The CV ylindrial method is used in our ylindrial ALE ode, and onservatively ooperates with the rest of the ALE algorithm. 4. Mesh Smoothing Tehniques The mesh smoothing is onsidered from a purely geometrial point of view. In the ylindrial geometry, we use the same smoothing tehniques, as desribed in setion 3. for Cartesian geometry, there is almost no hange needed. During the smoothing proess, the Cartesian ell volumes and ell enters are used, so that smoothing the same mesh in both geometries provides the same final mesh. The only point, where the mesh smoothing in ylindrial geometry is different from the Cartesian one, is the r = boundary ondition. In ylindrial geometry, the orresponding boundary nodes are allowed to move only along the z axis, they annot move in r diretion. The rest of the smoothing methods remains unhanged. 4.3 Conservative Remapping Algorithm For onservative ooperation of the remapping stage with the Lagrangian solver, we have to use the same ylindrial geometry, as desribed in setion 4... Let us go through all three parts of the remapping method (reonstrution, integration, and repair, and point to the hanges of the algorithm ompared with the original Cartesian method Pieewise Linear Reonstrution There is no need to hange the reonstrution proess (3.9 by minimization the error funtional ombined with the Barth-Jespersen limiter (3.96, (3.95, this method does not inlude any integration. 53

56 4.3. Numerial Integration Exat The derivation of the exat integration proess is straightforward. As in Cartesian geometry, we interset the new ell with all original ells in its neighborhood. We denote the intersetion by the symbol P. Analogially to the formula (3.98, we ompute the mass of this intersetion region as ( ( g = g r dr dz + r r dr dz r r dr dz G P P + r ( ( g z P P z r dr dz z P P r dr dz, (4.35 where the integrals of r, r, and r z are redued by the Green theorem to the boundary integrals, and evaluated from the oordinates of the region verties, as shown in setion 4... For ompleteness, we present here the final formulas for the required integrals over a general polygon P (ell, subzone, or as in this ase, ell intersetions in the ylindrial geometry r dr dz = (z z ( r 6 + r + r r (4.36 P P P r dr dz = z r dr dz = 4 e P e P e P (z z ( r + r (r + r (4.37 (z z ( r (3z + z + r (3z + z + r r (z + z, (4.38 where the verties of the edge e = [e,e ] have oordinates e = (r,z and e = (r,z. This is the only differene from the Cartesian geometry, the rest of the method remains unhanged, and all required numerial properties are satisfied Numerial Integration Swept As in the exat integration proess, the swept region integration method follows the Cartesian algorithm presented in setion For eah edge e of ell, we onstrut swept region used for mass exhange between the ell and the orresponding neighbor. As in the Cartesian ase, the swept mass δg e is obtained by integration of the reonstruted funtion from ell, in whih more of the swept region lies (3., whih is given by the sign of the swept region volume. This formula an be written in ylindrial oordinates in the form δg e = g (r,zr dr dz δ ( e ( ( g g = g r z ( ( g r r δ e r r dr dz + z ( g z δ e r dr dz δ e z r dr dz, where all the needed ylindrial integrals are evaluated in the previous subsetion by formulas (4.36, (4.37, and (4.38. The rest of the swept region integration routine remains unhanged, as well as all properties of this method. And analogially as in the Cartesian ase, the ylindrial remapping method is an approximate integration proess whih may ause the loal-bounds to be violated. Thus, the additional onservative repair stage must be performed to orret it. 54

57 4.3.4 Repair There is absolutely no need to hange anything in the repair proess presented in setion in ylindrial oordinates. During this stage, we are not performing any integration, we are just onservatively moving masses among ells in some neighborhood. We just have to note that, of ourse, for reomputation of masses to densities (a vie versa in ylindrial oordinates, the ylindrial ell volumes (4.7 must be used. 55

58 Chapter 5 Properties of Numerial ALE Method In this short hapter, we omment major properties of the desribed ALE algorithm and demonstrate them on a set of seleted well known fluid dynamis tests. Both Cartesian and ylindrial geometries are disussed in this setion. We also point here to possible violations of the required properties, whih may appear in some situations. 5. Conservativity The ondition of onservativity is naturally satisfied by the presented ALE algorithm. We disuss this property separately for the Lagrangian solver and for the remapping proess. The Lagrangian solver naturally satisfies the onservation of mass, masses in all the omputational ells remain unhanged during the Lagrangian omputation. Conerning the onservation of momentum and energy, their onservation strongly depend on the fluid movement on the boundary of the omputational domain. If the omputational domain does not hange during the simulation, and no shok waves, or other types of disontinuities appear on the boundary, the Lagrangian solver is onservative. For more details on the onservation issues of the Lagrangian method, see [8]. On the other hand, if the omputational domain hanges, the fores (and thus momentum and energy are applied to the deformation of the boundary, whih modifies the total momentum (energy, and they annot be preserved any more. Other possibility, how the onservation onditions an be violated, is adding some outer fore to the total nodal fores, for example the gravitation fore in the Rayleigh-Taylor instability problem [4, 5]. Due to this outer fore, the veloities are inreased, whih naturally affets the total momentum and energy in the domain. Let us onlude that for zero outer fore, stati omputational domain, and no disontinuous waves on the boundary, the Lagrangian step is onservative for all onservative quantities, and it is always onservative for mass. Conerning the remapping proess, it is applied to eah onservative quantity onseutively. We have already disussed in setion 3.3.6, that the remapping proess is onservative for all onservative quantities. This statement is orret, if the omputational domain does not hange during the mesh smoothing proess. For simple omputational domains, suh as square, the boundary nodes during smoothing only move along the boundary, the omputational domain does not hange its shape or volume, and the remapping method is exatly onservative. This is aused due to the fat, that masses are interhanged between neighboring ells via the swept masses fluxes and the available masses in the repair stage, but no mass is reated or removed. Unfortunately, it is not lear how to perform mesh smoothing on a general boundary, thus it may happen, that for the urved boundary, the smoothed mesh has different volume than the original one. Of ourse, if the volume of the domain is different, the mass is different too. The example of suh situation is shown in Figure 5.. It shows the Noh problem in D one of the lassial fluid dynamial problems, whih is a rare ase, when its analytial solution is known. The initial onditions are following. The fluid of density ρ = and zero pressure (p = 6 to avoid numerial problems is moving with the veloity w = towards the origin. The irular shok wave is reated, and 56

59 (a (b Figure 5.: The Noh problem omputed with purely Lagrangian method (a, and the simulation by omplete ALE method (b in the Cartesian oordinates. The density of fluid, and the omputational mesh are shown. spreads with the veloity /3 out of the origin. Inside the irular shok region, the fluid is at rest, and has the density ρ = 6. Outside of the irular region, the fluid is still moving with the initial veloity and has density ρ = + t/ x + y. In Figure 5., the solution omputed by the pure Lagrangian, and omplete ALE methods at time t =. is shown. Computational mesh with ells, and bulk artifiial visosity (3.36 was used for the simulations. The solution is omparable with the Eulerian simulations presented in [49]. For the purely Lagrangian method, the total mass is exatly onservative, and the total energy is different by.7% (due to movement of the boundary. For the omplete ALE method, where mesh smoothing/quantity remapping is applied after every Lagrangian steps (about 5 timesteps are needed till time t =, the mass onservativity is violated by.5% (due to urvature of the boundary, and the total energy by.4% (due to the ombination of both effets. The error of the total momentum is lose to the round off error in both ases thanks to the symmetry. As one an see, the umulation of the error during the omplete simulation due to the remapping non-onservativity on the boundary is lower than.5% for all quantities for this partiular example, and an be onsidered to be small enough. In the ylindrial oordinates, the situation with onservativity of the ALE algorithm and possible problems is ompletely the same, and does not require further desription. 5. Linearity Preservation, Order of Auray The linearity preservation ondition is satisfied in the Cartesian oordinates in the Lagrangian step, if the initial density values are set in the enters of the omputational ells. We demonstrate this on a linear density problem, where the fluid density in the omputational ells is initialized as a value of the following linear funtion ρ(x,y = x + y (5. in the ell enters and fluid pressure is set to p =. The initial mesh of ells is a randomly distorted square mesh in the omputational domain Ω =, moving with veloity w = (,. In Figure 5. (a, the purely Lagrangian simulation till time t = is shown. The mesh is exatly the same 57

60 (a (b ( 3 3 Figure 5.: Linear density problem: (a solution by purely Lagrangian method; (b remapped from the original randomly distorted mesh over a series of onsequently smoothing meshes; ( solution by the omplete ALE algorithm. as the initial one, it just moved with the speified veloity to the new loation. The numerial error is zero up to the round off error. The remapping proess is linearity preserving in the ase, that the initial density values are set in the enters of the ells subzones, and ell density is omputed from the subzonal quantities. The situation after 3 remap steps is shown in Figure 5. (b. The final omputational mesh is almost regular, and the numerial error is zero up to the round off error. The simulation by the omplete ALE method is presented in Figure 5. (, where the mesh smoothing/quantity remapping step was performed after eah Lagrangian step. In this ase, the linearity preserving ondition is not satisfied. On the other hand, the linearity preservation ondition is not required by the omplete ALE algorithm, only the seond order of auray is desired. For the presented linear density problem, the numerial error onverges with the seond order of auray for refining meshes. In the ylindrial geometry, one an just onsider the linearity preservation property in z diretion. During the movement in the r diretion in the Lagrangian step, the funtion profile in the r diretion is not preserved due to the geometry. 5.3 Numerial Tests In the previous setions 5., 5., we have presented two numerial examples of fluid dynamial simulations, the Noh problem and the linear density problem. Let us note, that both problems an be omputed by purely Lagrangian method, the omplete ALE results are presented here for demonstration of its properties. Let us return to the Noh problem and present the orresponding ylindrial results. Then, we present one more lassial test here, the Rayleigh-Taylor instability. Figure 5. presents the Noh problem omputed by the purely Lagrangian method (a, and the omplete ALE algorithm (b in the Cartesian geometry. The ALE algorithm does not affet the Lagrangian grid nor the solution, the density profiles are similar, and no signifiant diffusion appears. The Noh problem is diffiult for Lagrangian-type methods onsiderable density gap appears in the entral region. This gap is smaller in the ase of ALE omputation, it is partially orreted by the Eulerian mass flux between ells. The numerial errors of the omputations are similar, L = 5.% (L max = 79% for the pure Lagrangian and L = 4.7% (L max = 79% for the simulation by the omplete ALE method. The Noh problem simulation by our ALE method is desribed also in [5]. In the ylindrial oordinates, the D Noh problem an be simulated as a simple D problem in r diretion. In Figure 5.3, one an ompare plots of density omputed by both methods in r diretion in 58

61 (a (b Figure 5.3: The ylindrial Noh problem omputed by purely Lagrangian method (a, and the simulation by omplete ALE method (b. The density of fluid in the omputational ell enters is shown. time t =. Minor dissipativity in the shok region, and the region around the entral gap, an be seen, but in general, the mesh smoothing proess did not affet the solution signifiantly. Also, the speed of the shok wave propagation is orret, whih is often violated by the inorretly onstruted numerial methods. The last numerial example whih we present here, is the Rayleigh-Taylor instability (RTI simulation in the Cartesian geometry. The initial onditions onsist of two fluids in rest lying on eah other, the heavier one with density ρ = above a lightweight one with ρ =. The pressure is hydrostati, and the fluid interfae is disturbed a little by a sine like profile to start the instability evolution. The initial density on a 7 ells omputational mesh is shown in Figure 5.4 (a. This problem requires adding of the gravitation fore in the form Fg y l = g m l (5. in the vertial diretion y to eah total orner fore of subzone l of ell. Here, g denotes the gravitation onstant g =.. The gravitation and hydrodynamial pressure fores should keep the fluid interfae in rest if it would be horizontally flat. Due to the interfae perturbation, the equilibrium is violated, and the instability starts to evolve. The heavier upper fluid starts to stream down to the lighter one, and on the other hand, the lighter one starts to move up. We performed several Rayleigh-Taylor instability (RTI tests on a omputational mesh with 7 ells, the results in time t = 7. are presented in Figure 5.4 (b, (, and (d. The first test in Figure 5.4 (b is obtained by the appliation of the purely Lagrangian method with a high merit fator f m =.. For more details on the merit fator, see setion The high merit fator auses full influene of the subzonal pressure fore, whih prevents the mesh ells to perform rotational motions. This is usually orret, but in the RTI problem, the ells lose to the fluids interfae must move this way to evolve the instability. Thus, in this ase, the strong subzonal pressure fore does not allow the instability to be reated. On the other hand, if the merit fator is low f M =. as in Figure 5.4 (, the ells are allowed to deform more. Beause we use purely Lagrangian method, no mixing of the fluids happens, and the rotated ells are very long and narrow. The instability does not evolve again, beause the fluids interfae follows the edges of these long ells. In the last plot in Figure 5.4 (d, the appliation of the omplete ALE algorithm with the low merit fator and mesh smoothing/quantity remapping proess after every Lagrangian steps is shown. Some diffusion is added along the fluids interfae, whih represents their mixing. Due to the smoothing proesses, all omputational ells have reasonable shapes now, and the 59

62 (a (b ( (d Figure 5.4: Density distribution of the Rayleigh-Taylor instability problem: (a initial data; (b Lagrangian simulation with high merit fator f M =.; ( Lagrangian simulation with low merit fator f M =. (d ALE simulation with low merit fator f M =. and smoothing/rezoning stage after eah Lagrangian steps. All (b, (, (d simulations at time t = 7.. instability starts to evolve into the mushroom shape. The simulation by the omplete ALE method an ontinue without any problems till time t = 8.5, whih is usually the final time of the RTI simulations. We used the shorter time, beause the low merit Lagrangian simulation ( fails shortly after t = 7.. For further details about the ALE simulations of the RTI problem, see [4, 5]. In the ase of the Noh problem, we have shown that the ALE method does not affet the positive properties of the Lagrangian mesh motion, both methods produe similar solutions not degraded by the used mesh smoothing tehniques. On the other hand, in the ase of the Rayleigh-Taylor instability problem, the Lagrangian method in our ase was not able to produe reasonable results, whih ALE method produes. This main advantage of the ALE approah is even more visible for the high veloity impat problems simulations, whih are presented in hapter 7, where the purely Lagrangian approah fails due to severe mesh distortions, and no reasonable results are obtained without the omplete ALE tehnique. 6