High Order Accurate Curved-element Implementation in OpenFOAM for Discontinuous Galerkin Method

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1 , htt://dx.doi.org/ /astl High Order Accurate Curved-element Imlementation in OenFOAM for Discontinuous Galerkin Method Li-Yang Xu 1, Shuai Ye 1, Yong-Quan Feng 1, Hao Li 1, Xin-Hai Xu 1, Xiao-Guang Ren 1,* 1 State key Laboratory of High Performance Comuting ational University of Defense Technology College of Comuter ational University of Defense Technology Changsha, China {xuliyang08, yeshuai09, fengyongquan12, lihao, xuxinhai, renxiaoguang}@nudt.edu.cn Abstract. A high order accurate discontinuous galerkin method based curvedelement imlementation in OenFOAM is resented in this work. The degree of freedom oints on origin straight side mesh are maed to curvilinear edges by finding nearest oints on boundary. Then the deformation are transorted to the inner oints by blended method. Based on the run-time dynamic code of OenFOAM, we design a convenient interface for users to configure the arameterized function of curvilinear boundary. The curved-element imlementation is alied to the simulation of vacuum Maxwell equations in metallic air-filled cavity. Results show that the curved-element has significantly imroved the mesh quality. High order numerical convergence rate has been achieved. Keywords: curved-element, high order, discontinuous galerkin, OenFOAM 1 Introduction In the last few decades, comutational fluid dynamics (CFD) has been more and more imortant in both scientific research[1][2] and industrial manufacture. At the meantime, significant rogress has been made in the research of high order numerical methods[3]. Comaring to low order methods, high order methods can rovide more accurate simulation results in the same comutational costs. Discontinuous Galerkin Method (DGM) is one of the most oular and widely used high order method. DGM was firstly roosed by Reed and Hill in the 1970s to solve the neutron transort roblem[4]. Cockburn and Shu introduced Runge-Kutta time steing to DGM and succeed in simulation of conservation laws system[5] [7]. Based on their creative efforts, a serial of work has extended the DGM to most of the CFD area. * State Key Laboratory of High Performance Comuting, ational University of Defense Technology, Changsha, Hunan, China. renxiaoguang@nudt.edu.cn ISS: ASTL Coyright 2016 SERSC

2 The released DGM software, such as Deal.II [8] and Fenics [9], have comlex user interface and weak suort for curved boundary. Better designed DGM software is an urgent task for CFD. OenFOAM[10] is a mature oen source CFD software based on finite volume method. Due to its excellent frame work design and convenient interfaces, it gained great reutation in CFD community. However, the numerical discretization of OenFOAM is 2rd order at most. Out grou is trying to introduce DGM in OenFOAM to realize high order numerical simulation, and rovide convenient user interface consistent to OenFOAM. The main work of this aer include: 1. Imlementation of high order accurate curved-element in OenFOAM for DGM; 2. Design of convenient interface to configure arameterized curve function based on run-time dynamic code; 3. Simulation of Maxwell equations in a metallic air-filled cavity to validate the curved-element and convergence rate. The rest of the aer is organized as follows. Section 2 briefly introduce the main feature of DGM and the reresentation of curved boundary. The algorithm to calculate the curved-element and software design are described in section 3. Section 4 demonstrate the effects of curved element in geometry and numerical accuracy with the simulation of Maxwell equation. Conclusions are in Section 5. 2 Discontinuous Galerkin Method We adot the nodal DGM originally roosed by Hesthaven and Warburton[11]. Here we briefly introduce the numerical scheme. Taking u as the unknowns, Eq. (1) can be rewritten as conservation law in Eq. (2). u F ( u ) 0, x t, t 0, 0 u u t, 0. The comutational domain is divided into a collection of non-overlaing elements. (2) i1 Then, we multily the conservation law by a test function each element i. By using the green theory, we finally get i. (3) and integrate over tuds F( u) nd F( u) ds 0. (4) i i i Where n reresents the normal vector on i. The unknowns u can be described as the linear combination of nodal Lagrange basis{} i i1 or modal basis{ i} j 1, where 48 Coyright 2016 SERSC

3 u u ( x) c ( r), k 1,2,3,4, (5) k k, i k, i k, j k, j i0 j0 3 Methodology This section firstly introduces the algorithm to calculate the DOF location of curvedelement. To rovide a convenient interface for users to describe and configure the curved boundary, we design a dictionary-based interface in the second art. Consider a curved triangle element in domain with boundary defined by arameterized function. In most CFD software, edges in mesh are all straight as shown in the left subfigure of Fig. 1. There is a ga between the straight edges with the analytical curvilinear edge (the dotted line). The straight sided mesh will introduce error to the DGM discretization. To achieve consistent high order accuracy, the DOF locations should be redistributed according to the curved boundary. S( a, b) Fig. 1. Left: Original straight side element; Middle: One face made curvilinear; Right: Deformation Blended into inner oints. Firstly, we focus on the DOF oints on the boundary face and trying to ma them to the curve S( a, b ). The deformation of DOF oints on the face can be found by solving where x M 1 2 min xm S( a, b ), (7) ( ab, ) 2 is the DOF location on the origin straight side mesh. We use ewton-gauss iteration to solve this nonlinear roblem, and obtain the maed oints on the curve. With the maed oints, the curve now can be reresented by the basis in n th order form analytically as Eqn. (8). The difference ( x ) between x M and x S reresents the deformation of facial DOF oints. ( ) S x x ( x ). (8) i0 S, i k, i x S Coyright 2016 SERSC 49

4 Secondly, we focus on the inner DOF ints in the curved-element. The inner DOF should be shifted consisted with facial DOF oints. The deformation is transorted to inner oints by blended rule to remain the LGL roerty. The method introduced in the revious subsection can deal with any kind of curve, but the arameterized function should be rovided as the recondition. A simle way is to write the function in code directly, which is inflexible and the code has to be recomiled if the curve changed. OenFOAM includes the caability to comile, load and execute C++ code at run-time. The comonent called codestream can be used to generate user defined interface from configuration dictionary. Inheriting from this comonent, we imlement an interface to rovide a swift way to configure custom curvilinear function. S( a, b) 4 Exeriments and Results To demonstrate the effects of curved-element, we simulate the vacuum Maxwell equations in a metallic air-filled cavity. A series of meshes with different size are used to test the order of convergence. Boundary of the comutational domain is a circle h /2and h /4 with radius 1.0. Three meshes with size h, are generated by Gambit. Curved boundaries are aroximated by iecewise straight edges. DGM curved elements mainly take effects on two asects, the geometry of boundary elements and numerical convergence rate. After alied curved element, the DOF oints in the outermost elements are shifted according to the arameterized curvilinear function. Fig. 2 illustrate the details of mesh deformation. It is clear that the curved elements have significantly imroved the mesh quality. The DOF oints are located almost exactly on the analytical curvilinear boundary. Fig. 2. Deformation of curved element on boundary faces. Left: original straight side mesh; Right: mesh with 4 th order curved-elements. We run the case with a transient solver based on OenFOAM and DGM. Simulation results obtained from mesh with size h /2 and 4 th order at time= 0.5 is showed in Fig Coyright 2016 SERSC

5 Fig. 3. Simulation result of electric field E with 4 th order curved-elements at time=0.5, on the mesh with size. h /2 To check the convergence order, we run the case in a sequence of meshes and different base orders ranging from 1 to 6. Table. 1 lists the L 2 error and convergence rate simulated by DGM without curved-elements, while Table.2 lists the results by DGM with curved-elements. Though the order of basis increasing from 1 to 6, the convergence rate remains 2. Contrast to this, Table.2 shows that simulation with curved elements can reach the theoretical convergence rate. Table 1. Results without curved-elements. L 2 error and convergence rate of electric field E obtained from the simulation based on different basis order and mesh size. The mesh can not get convergenced when equals 5 or 6. h h /2 h / e e e e e e e e e e e e e e e e h /4 Convergence rate Table 2. Results with curved-elements. L 2 error and convergence rate of electric field E obtained from the simulation based on different basis order and mesh size. h h /2 h / e e e e e e e e e e e e e e e e e e Convergence rate Coyright 2016 SERSC 51

6 5 Conclusion A high order accurate DGM based curved-element has been imlemented in OenFOAM. The curved-element has significantly imroved the mesh quality for curvilinear boundary. High order accurate convergence can be achieved in DGM simulation. Describing the curvilinear boundary function is simle and direct. With the suort of run-time dynamic code stream, arameterized functions are set in the boundary control dictionary. Though the case in exeriment is simle, this imlementation can be used for much more comlex cases. Extending to 3-dimension case is natural. o need to make any change, this method can be alied to arallel comuting. But the decomosing of curvilinear boundary should be treated secially. Acknowledgments. The authors declare that there is no conflict of interest regarding the ublication of this aer. The authors would like to thank the ational atural Science Foundation of China (Grant no ) and the Oen fund from State Key Laboratory of High Performance Comuting (no and ) for funding. References 1. Yang, W.-J., Yi, W., Ren, X.-G., Xu, L.-Y., Xu, X.-H., Yuan, X.-F., Toward large scale arallel comuter simulation of viscoelastic fluid flow: A study of benchmark flow roblems, J. onnewton. Fluid Mech., Oct Zou, S., Yuan, X., Yang, X., Yi, W., Xu, X.: An integrated lattice Boltzmann and finite volume method for the simulation of viscoelastic fluid flows, J. onnewton. Fluid Mech., vol. 211, , Cheng, J., Shu, C.: High order schemes for CFD: A review, Comut. Phys.,. 1 50, Reed, W. H., Hill, T. R.: Triangularmesh methodsfor the neutrontransortequation, Los Alamos Re. LA-UR , Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local rojection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comut. Phys., vol. 84, no. 1, , Cockburn, B.: An Introduction to the Discontinuous Galerkin Method for Convection - Dominated Problems, in Advanced umerical Aroximation of onlinear Hyerbolic Equations, 1998, Cockburn, B., Shu, C.: Runge Kutta Discontinuous Galerkin Methods for Convection- Dominated Problems, J. Sci. Comut., vol. 16, no. 3,. 89, Bangerth, W., Hartmann, R., Kanschat, G.: deal.ii --- a General Purose Object Oriented Finite Element Library, ACM Trans. Math. Softw., vol. 33, no. 4,. 24, Barth, T. J., Roose, D.: The FEniCS Book. Sringer. 10. Jasak, H.: Jemcov, A., Tukovic, Z.: OenFOAM : A C ++ Library for Comlex Physics Simulations, Int. Work. Couled Methods umer. Dyn., vol. m,. 1 20, Hesthaven, J. S., Warburton, T.: odal Discontinuous Galerkin Methods, vol Coyright 2016 SERSC

Efficient Parallel Multi-Copy Simulation of the Hybrid Atomistic-Continuum Simulation Based on Geometric Coupling

, pp.384-389 http://dx.doi.org/10.14257/astl.2016.139.77 Efficient Parallel Multi-Copy Simulation of the Hybrid Atomistic-Continuum Simulation Based on Geometric Coupling Qian Wang 1,2, Xin-Hai Xu 1,2,*,