Discontinuous Galerkin Finite Element Methods for Shallow Water Flow: Developing a Computational Infrastructure for Mixed Element Meshes

Transcription

1 Discontinuous Galerkin Finite Element Methods for Shallow Water Flow: Developing a Computational Infrastructure for Mixed Element Meshes A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Ashley L. Maggi, B.S. Graduate Program in Civil Engineering The Ohio State University 2011 Master s Examination Committee: Ethan J. Kubatko, Advisor Gil Bohrer Shive Chaturvedi

2 c Copyright by Ashley L. Maggi 2011

3 ABSTRACT Discontinuous Galerkin finite element methods (DG FEM) for the shallow water equations using mixed element meshes that consist of triangular and quadrilateral elements in two dimensions and triangular prisms and hexahedra in three dimensions are developed, implemented, and tested. The main motivation behind this work is to gain more insight on whether the use of quadrilateral/hexahedral elements improves the efficiency of DG methods in a setting in which two(adjacent) triangular/triangular prism elements are merged to form a single quadrilateral/hexahedral element. The elements that are used in this study are constructed from a set of orthogonal, modal basis functions formed from products of Legendre and Jacobi polynomials. Given the fact that DG methods do not require continuity of the approximate solution between elements, quadrilateral and hexahedral element basis functions may be developed that exclude the usual cross-terms that are present in standard C 0 elements, e.g., a linear quadrilateral element may be used instead of a bilinear quadrilateral element. The performance of the developed DG methods on triangular meshes and quadrilateral meshes of arbitrary polynomial order p is evaluated in terms of accuracy and computational time on a set of analytic test cases for the linear shallow water equations. The numerical results provide evidence that there is substantial benefit in using quadrilateral elements, and it is expected that the hexahedral elements will offer similar computational savings over the triangular prism elements. ii

4 This work also focuses on improving the computational efficiency of the existing triangular prism elements. FEM is rapidly progressing in multi dimensional and multi regional domains in which a crucial point in computing the FEM solutions is the evaluation of the domain integrals arising over the master element. Previously, nonproduct rules have not been developed specifically for the triangular prism, as they have for many other commonly used shapes. Shape specific nonproduct rules for numerical integration over the triangular prism domain are derived in this work, which result in, on average, a 27% computational savings over each element for each time step of the problem. This research has laid the groundwork for a robust computational infrastructure in the context of DG FEM that will significantly cut computational cost by applying fast and efficient state of the art algorithms. The promising results provide motivation for future model development within the framework of a mixed element approach as well as the derivation of higher order numerical integration rules for triangular prism domains. iii

5 In loving memory of my papa iv

6 ACKNOWLEDGMENTS I would like to express my sincerest gratitude to my advisor, Professor Ethan Kubatko, for his encouragement to pursue a graduate degree and the wonderful opportunity he has provided me. In addition, I am grateful of his limitless guidance, patience and support throughout the course of this research. I would also like to thank my family for their love and unconditional support to pursue my dreams, whatever they may be. v

7 VITA June 2004 Thomas Worthington High School June 2010 B.S. Civil Engineering, The Ohio State University vi

8 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita List of Tables List of Figures ii iv v vi ix xi Chapters: 1. Introduction Motivation & Objectives Prior Work Thesis Organization Mathematical Preliminaries Governing Equations Discontinuous Galerkin Formulation The Elements D Elements Triangular Elements Rectangular Elements D Elements vii

9 3.2.1 Triangular Prism Element Rectangular Hexahedron Elements Element Basis Function Subroutine Numerical Quadrature and Cubature Triangle Quadrature Rules Square Quadrature Rules Triangular Prism Cubature Rules Method of Construction Symmetry groups Solution of System of Equations Some Cubature Formulas for Triangular Prisms Cube Cubature Rules Quadrature Subroutine Numerical Testing & Results Analytic Test Case Results Conclusions and Future Work Bibliography viii

10 LIST OF TABLES Table Page 4.1 Number of Quadrature Points: Triangle Number of Quadrature Points: Square Number of Cubature Points: Triangular Prism Distribution of points and unknowns for symmetry groups Triangular Prism Cubature Rule: p = Triangular Prism Cubature Rule: p = 2 a Triangular Prism Cubature Rule: p = 2 b Triangular Prism Cubature Rule: p = 3 a Triangular Prism Cubature Rule: p = 3 b Triangular Prism Cubature Rule: p = Triangular Prism Cubature Rule: p = Number of Cubature Points: Cube Test Case Data Mesh Data L Error ix

11 5.4 L 2 Error Rate of h-convergence Ratio of CPU Time x

12 LIST OF FIGURES Figure Page 1.1 Unstructured Mesh Representation Structured Mesh Representation Mixed Element Mesh Representation Shallow Water Assumption Shallow Water Diagram Finite Element Partition of Domain Piecewise Discontinuous Subspace Bilinear vs. Linear Functions Master Triangular Element Master Triangular Element Transformation Mapping of Triangular Master Element to Quadrilateral Triangular Hierarchical Basis Functions Master Rectangular Element Master Rectangular Element Transformation Rectangular Hierarchical Basis Functions xi

13 3.9 Staircase Boundary Sigma-Coordinate Transformation Master Triangular Prism Element Mapping of Triangular Prism Element to Hexahedral Master Cubic Element Pseudo Code of Orthogonal basis.f Quadrature rules for the triangle Quadrature rules for the square Reference Triangular Prism Symmetry groups used for the triangular prism Quadrature rules for the triangular prism Quadrature rules for the cube Schematic of quadrature subroutine Test Case Domain Test Case Bathymetry Test Case for Simulation h-convergence: Quadrilateral Elements h-convergence: Triangular Elements h-convergence: Both Element Types p-convergence: Surface Elevation p-convergence: Velocity xii

14 5.9 Ratio of CPU Time xiii

15 CHAPTER 1 INTRODUCTION The development of computational tools that can provide reliable and efficient approximations to solutions of partial differential equations (PDEs) is an important focus of the scientific and engineering community. These types of computational models are used to simulate a wide range of physical phenomena such as flow dynamics in the atmosphere and ocean, the deformation of structures, the advection of contaminants in natural systems, etc. and with continuous advances in computational power, applied mathematics, and the availability of physical data (e.g., lidar based data), they are being applied to problems of greater and greater complexity. While this situation creates an opportunity to solve problems that were thought to be intractable only a few years ago, it also creates a need to continually update the algorithms and tools used by these models so that they can continue to provide accurate and reliable data in an efficient manner. This thesis specifically addresses the continued development of a computational tool that is used for solving time dependent free surface flow and transport problems in two and three dimensions. The shallow water equations, a set of hyperbolic PDEs, are used to model free surface flow in the deep ocean, coastal ocean, estuaries, rivers, open channels, and coastal floodplains. Many flow processes are described by these 1

16 equations including tides, the propagation of flood waves in rivers, tsunami waves, and storm surges associated with hurricanes. The last two examples are indicative of the kinds of applications that require efficient codes and quick turn around time in performing simulations so that crucial information can be provided to emergency managers and first time responders as storms approach landfall. In many practical applications, like the one shown in Figure 1.1, the physical domains over which the shallow water equations are solved are geometrically complex, which is introduced by the complicated vertical and horizontal boundaries arising along the coastline and the bottom surface. Additionally, there is generally a large range of scales present in the problems, from the relatively quiet waters of the deep ocean to the more dynamic flows found in the coastal area. The geometry and the range of scales of the given problem must be adequately represented by a so-called computational mesh in order for the computational model to provide accurate solutions. The computational mesh consists of a collection of nodes and/or elements over which problem data is specified and approximate solutions are sought. Computational meshes fall into one of two main categories: 1. Unstructured meshes, which spatially vary the density or resolution of nodes and elements throughout the domain. These types of meshes can be used to easily fit the complicated boundaries as well as allow for varying element size to capture the different scales of the problem efficiently. Finite element methods (FEM) make use of unstructured meshes and thus are particularly advantageous for the types of problems considered here. Figure 1.1 shows an example of part of an unstructured mesh that is used for simulating hurricane storm surge in 2

17 Southern Louisiana. The contour colors indicate element size, which varies from 5 km in the deep ocean to less than 50 m nearshore. 2. Structured meshes, which utilize an equal density (resolution), or spatially nonvarying, distribution of nodes throughout the domain. Finite difference methods typically use these types of meshes. With structured meshes it is not as easy to fit complicated boundaries while simultaneously capturing the varying scales of the problem in an efficient way. The structured mesh shown in Figure 1.2 demonstrates, in a simple way, how difficulties can arise in trying to fit complicated boundaries using a fairly coarse structured mesh, which results in a staircase effect along the boundary. This complicated boundary could be resolved using a much finer structured mesh, however, this would result in a large computational overhead as the larger scales of the problem would no longer be economically captured. Of course both of the above mentioned mesh types have their advantages and disadvantages. While unstructured meshes can easily fit the complicated boundaries and allow for varying element size, they can be very costly and time consuming to generate as well as present challenges during numerical implementation. On the other hand, structured meshes are much easier to create and from a numerical perspective are simpler and more efficient to implement. These facts lead to the investigation of creating a computational infrastructure in the context of FEM which utilizes the advantages of both mesh types specifically, the consideration of a mixed element mesh approach. Mixed element meshes attempt to tailor the amount of work involved to obtain approximations to the problem being solved, using, for example, a having a finer unstructured mesh where the solution and/or geometry varies rapidly and a 3

18 Figure 1.1: This particular unstructured mesh is of Southern Louisiana and is part of a larger mesh used in hurricane simulations. Figure 1.2: Structured mesh representation of complicated boundary(solid dark line). 4

19 Figure 1.3: Mixed Element Mesh Representation coarser simple structured mesh where the solution and/or geometry does not vary as much. This thesis addresses implementation of mixed element meshes for a specific variant of FEM called the discontinuous Galerkin (DG) method. Briefly, the DG method differs from the standard, continuous Galerkin FEM in that continuity constraints are not imposed on quantities on the interelement boundaries, thus resulting in an approximate solution which is composed of piecewise continuous functions with possible jump discontinuities along the element edges. This thesis contains a mix of mathematical theory and numerical implementation detail, all directed toward obtaining more efficient approximations to the representative shallow water equations utilizing DG FEM. 5

20 1.1 Motivation & Objectives The specific form of the discrete equations that arise from the DG formulation depend on the particular element type employed. The current version of the code uses triangular elements in 2D and triangular prism elements in 3D. A profile of the code revealed that nearly two-thirds of the computational time was spent on element calculations. This obviously indicates that the particular type of element type chosen greatly affects the efficiency of the code. This fact serves as one of the main motivations for the investigation of different element types and the proposed mixed element mesh approach. To this end, this thesis work is focused on three main tasks: 1. Implement new quadrilateral (2D) and hexahedral (3D) elements and compare their efficiency to the existing triangular and triangular prism elements. 2. Modifythecodetoaccommodatetheuseofbothelementtypesinasinglemesh. 3. Work on improving the efficiency of the existing triangular prism elements. 1.2 Prior Work The use of mixed element meshes within the framework of DG methods has been limited, and to the author s knowledge nonexistent in the application of models involving the shallow water equations. There has been work done by [2], in which a numerical analysis is extended from traditionally studied and used triangular element meshes to quadrilateral element meshes in the context of viscoelastic flow. After a detailed mathematical application of DG methods to the transport equation 6

21 on a quadrilateral mesh, the work concludes that the error in the approximation is sufficiently small and converges in the energy norm for the unknowns. In the discipline of so-called non-conforming meshes (i.e., meshes where the location of the neighboring element vertices do not necessarily match up), a mixed element mesh approach has been taken by [16] to compute 2D seismic wave propagation. The scheme was formulated to achieve the same approximation order in space and time and to avoid numerical inconsistencies due to non-conforming mesh transitions or the change of the element types. The performance of the new scheme was verified by numerical convergence tests and experiments with comparisons to independent reference solutions obtained by previous all triangular element schemes. The findings concluded that the scheme reduced the computational cost in the sense of memory and run-time requirements and the authors suggested similar behavior would be observed for an extension to three space dimensions. The most relevant work previously conducted in this specific area of research has been done by [28], in which a performance comparison of nodal DG methods on triangles and quadrilaterals was applied to hyperbolic conservation laws. The performance of the two element types was assessed by applying each mesh to a linear advecting rotating plume transport problem. The test cases concluded that the computation time needed for the quadrilateral elements was shorter than that for the triangular elements. The numerical results also revealed that the quadrilateral elements yield higher computational efficiency in terms of cost to achieve similar accuracy. The promising findings of [28] initiated in part, the motivation for this work. 7

22 1.3 Thesis Organization This thesis is organized as follows. Chapter 2 provides the background details of the mathematical model and introduces the governing equations for the physical processes we are trying to model. The numerical solution of these equations using the DG method is then discussed, and an outline of the DG formulation is provided. Following this, in Chapter 3, some specific implementation details of the elements chosen in both 2D and 3D are described. This includes their definition and transformation to a master element, the set of basis functions chosen to approximate the solution over each element, and a schematic of the subroutine implemented in the model. In Chapter 4, quadrature points and weights for numerically evaluating integrals over the elements are defined. Included in this chapter is the derivation of a set of novel rules for integrating over the triangular prism domain in three dimensions. Chapter 5 gives the results of the numerical testing performed with the modified code, while Chapter 6 draws some conclusions, gives the future directions of this work, and touches on some of the challenges that lie ahead. 8

23 CHAPTER 2 MATHEMATICAL PRELIMINARIES This chapter provides the basic information required to describe the implementation of the computational model described in later chapters. 2.1 Governing Equations The governing equations for shallow water hydrodynamics arise from the Navier- Stokes equations, which are the general equations for fluid motion, under the assumption that the vertical length scale of the problem is much smaller, or shallower, compared to the horizontal length scale. For example, if H is the depth of the ocean and L is a typical wave length, the shallow water assumption says H/L is much less than 1. In this case, the balance of momentum in the vertical z direction reduces to the hydrostatic approximation for the pressure, i.e., p = p 0 +ρg(ζ z). The remaining horizontal momentum equations, together with the conservation of mass or continuity equation, form what are sometimes referred to as the three dimensional shallow water equations. If we then integrate these equations over the depth, we arrive at the depth-integrated or depth-averaged shallow water equations, often 9

24 L H The shallow water assumption: H L << 1 Figure 2.1: Shallow Water Assumption just referred to as the shallow water equations ζ t + x (Hu) + y (Hv) = 0 t (Hu) + ( Hu ) x 2 g(h2 h 2 ) + y t (Hv) + x (Huv) + ( Hv ) y 2 g(h2 h 2 ) (Huv) = gζ h x + F x = gζ h y + F y. (2.1) This set of hyperbolic PDEs is begin solved for the height of the free surface, ζ, and the depth-integrated horizontal velocities, u and v over some domain Ω. The datum is typically set at the still water level, h is the bathymetric depth measured from this datum, H is total height of the water column, g is the acceleration due to gravity, and F x and F y account for additional terms such as Coriolis force, tidal potential force, and wind or wave radiation stress, that may be present. A schematic of the shallow water model is given in Figure 2.2. Like any set of differential equations, suitable boundary and initial conditions need to be specified. Typical boundary conditions include the specification of the free surface elevation and or the flow condition along the boundary of the domain, 10

25 Free surface ζ(x, y,t) Datum ζ = 0 u(x, y, t) h(x, y) H(x, y,t) Bottom boundary Figure 2.2: Shallow Water Diagram and the initial conditions are the given by the initial state of the free surface and the initial velocities. In solving hydrodynamic problems, sometimes we are only looking for the flow conditions, but often we use the flow to drive another process we want to model, i.e., contaminant transport, sedimentation, salinity, etc. These processes, as well as the above shallow water equations, all fall into the general form of time-dependent conservation laws where the time rate of change of some quantity is being balanced by the divergence of the flux of that quantity and some set of source or sink terms, s u i t + F i = s i. This is the preferred form of the equations for applying the DG method which will be presented in the section to follow. 11

26 Ω T h = {Ω e } Figure 2.3: Example of a finite element partition of a domain Ω 2.2 Discontinuous Galerkin Formulation As stated in the introduction, DG methods are finite element methods, which use a weak or variational form of the problem. To arrive at this form starting from the PDE model and a finite element partition T h of the domain Ω, first multiply by a sufficiently smooth test function v and integrate over each element, Ω e u t v dω + Ω e F v dω = Then the divergence term of Equation (2.2) is integrated by parts, Ω e s v dω. (2.2) Ω e u t v dω Ω e F v dω + Ω e F n v ds = Ω e s v dω, u,v W which removes the derivatives off of the flux function F and onto the test function v, while also creating a boundary flux term, which is essentially how the elements communicate in DG methods. The true solution u and the test function v belong to some space of functions W, but we look for an approximate solution that is in some subspace of W denoted 12

27 v h p = 1 p = 2 Figure 2.4: An example of piecewise discontinuous elements in the subspace P k (Ω e ) by W h, that is spanned by a basis of {φ}; that is, any function W in W h can be represented by a sum of the form W = w i φ i where {w i } is a set of coefficients and {φ i } is the basis of the space. We then approximate the true solution u by u h W h, that is N u u h = u i (t)φ i (x,y) i=1 For example, we could choose the subspace to be P k (Ω e ), the set of piecewise polynomials of degree k defined over each element, that is, W h = { } w : w Ωe P k (Ω e ) Ω e T h and then we replace the u and v by their approximations u h and v h in the weak form. This process leads to the discrete weak form of the problem which consists of a set of integral equations for each element, 13

28 for i = 1,2,...,N Ω e u h t φ idω Ω e T e Ω e (F φ i +sφ i )dω + Ω e F nφ i ds = 0, (2.3) the number of which N depends on the order of approximation that is being used; for example, linear, quadratic, etc. Thus, the problem is converted to a system of ordinary differential equations (ODEs) of the form M. u = L(u), M = Ω e φ 1 φ 1 dω e Ω e φ 2 φ 1 dω e... Ω e φ 1 φ 2 dω e Ω e φ 2 φ 2 dω e Ω e φ 1 φ N dω e Ω e φ 2 φ N dω e... Ω e φ N φ 1 dω e Ω e φ N φ 2 dω e. Ω e φ N φ N dω e,. u=. u 1. u 2.. u N where M is the mass matrix, L is the DG operator, and. u is the vector of time derivatives of the unknown coefficients u i. This system of ODEs is then solved in time using Runge-Kutta methods. 14

29 CHAPTER 3 THE ELEMENTS In this work, an element class is defined by three main concepts: (i) a geometric mapping or transformation from the physical element, Ω e, to a reference or master element, ˆΩ, (ii) a set of basis functions, {φi }, and their corresponding degrees of freedom, that define the subspace of the approximate solution over the element, and (iii) a set of quadrature (2D) or cubature (3D) points and weights for numerically evaluating the integrals over the element domain, i.e., the integrals appearing in the discrete weak form are replaced by a weighted sum of functions evaluations. The first two items are discussed in this Chapter, while the discussion of the numerical evaluation of the integrals is contained in Chapter 4. Without the constraint of maintaining C 0 continuity of the approximate solution between elements in the DG method, there exists a greater freedom in the choice of elements that may be used, especially in the choice of basis functions. This fact is particularly relevant in the case of the new quadrilateral and hexahedral elements that are implemented in this work. For example, in constructing continuous basis functions over quadrilateral elements for the standard continuous Galerkin FEM, it is necessary to use bipolynomial functions in order to maintain continuity between elements. This idea is illustrated in Figure (3), which compares the form of a bilinear 15

30 Bilinear function f(x, y) = a + bx + cy + dxy Linear function f(x, y) = a + bx + cy Figure 3.1: A comparison of bilinear (left) and a linear (right) functions defined over a quadrilateral element. and a linear function over a quadrilateral element. The bilinear function contains an additional quadratic cross term, xy, that is required to maintain continuity between elements. These extra cross terms that appear in the bipolynomial functions introduce additional degrees of freedom (and computational cost) and are solely required in order to generate continuous approximate solutions. They do not contribute to the asymptotic rate of convergence of the element interpolation error [3]. Therefore in this work, we focus on constructing quadrilateral and hexahedral elements that use simple polynomial basis functions instead of the usual bipolynomial functions used in standard FEM. This results in a set of simpler and more efficient quadrilateral and hexahedral elements. 16

31 Along these same lines, each element class discussed here also uses a set of orthogonal basis functions; that is, they have the property C ij if i = j φ i φ j dˆω = ˆΩ 0 if i j where C ij is a constant that varies by element type and degree. This results in a diagonal mass matrix, that is, M = C C C nn, (3.1) which requires no matrix inversion and results in increased computational efficiency. Additionally, each set of basis of functions that is used is also hierarchical, meaning that the set of basis functions of degree p is a subset of the set of basis functions of degree p + 1, which is a subset of the set of basis functions of degree p + 2, etc. This property is especially convenient and computationally efficient when implementing dynamic element order adaptation. Lagrange basis functions that are used in standard FEM do not possess this property D Elements We begin with a description of the 2D elements that have been implemented in the code namely, the existing triangular elements and the newly constructed rectangular elements. In each case, a description of the master element geometry, including relevant mappings or transformations, and the set of basis functions is described. 17

32 ξ 2 ( 1, 1) Edge 2 ( 1, 1) 3 Edge 1 ˆΩ 1 2 Edge 3 ξ 1 (1, 1) Figure 3.2: Master Triangular Element Triangular Elements Master Triangular Element A reference or master triangular element ˆΩ is defined using a local Cartesian coordinate system, (ξ 1,ξ 2 ) [ 1,1], as defined in Figure 3.2. The vertices of the master element are numbered in a counter-clockwise manner, and edge number i is defined as that edge opposite vertex i (circled numbers). A physical element Ω e in the (x,y) coordinates can be defined as the image of this master element under the affine transformation T e : ˆΩ Ωe defined by x = 1 [ (ξ1 ) +ξ 2 x1 ( ) 1+ξ 1 x2 ( ] ) 1+ξ 2 x3 2 y = 1 [ (ξ1 ) +ξ 2 y1 ( ) 1+ξ 1 y2 ( ] ) 1+ξ 2 y3 2 18

33 T e : (x, y) (ξ 1, ξ 2 ) (x 3, y 3 ) ξ 2 ( 1, 1) y Ω e (x 2, y 2 ) ˆΩ ξ 1 (x 1, y 1 ) ( 1, 1) (1, 1) x Figure 3.3: Master Triangular Element Transformation where (x 1,y 1 ), (x 2,y 2 ), and (x 3,y 3 ) are the physical coordinates of the vertices of Ω e, numbered locally in a counter-clockwise manner. The Jacobian of this transformation is J = (x, y) (ξ 1, ξ 2 ) = 1 [ x2 x 1, x 3 x 1 2 y 2 y 1, y 3 y 1 ] J = A( ) e 2, where A ( ) e is the area of Ω e. The inverse of the transformation T 1 e : Ω e ˆΩ is given by ξ 1 = 1 A e { (y3 y 1 ) [ x 1 2 ( x2 +x 3 ) ] + ( x 1 x 3 ) [ y 1 2 ( y2 +y 3 ) ]} ξ 2 = 1 A e { (y1 y 2 ) [ x 1 2 ( x2 +x 3 ) ] + ( x 2 x 1 ) [ y 1 2 ( y2 +y 3 ) ]}. 19

34 Given these transformations, the area integrals over the physcial element in (x, y) coordinates can be related to the area integrals over the master element in local coordinates (ξ 1,ξ 2 ). Specifically the relationship is given by Triangular Basis ( ) e f da = A Ω e 2 f dâ. ˆΩ An orthogonal basis for the triangle can be constructed as a generalized (or warped, see [7]) tensor product of the so-called principal functions defined as ψ (1) i = P (0,0) i (η 1 ), ψ (2) ij = ( 1 η2 2 ) i P (2i+1,0) j (η 2 ), where P (α,β) n is the n-th order Jacobi polynomial of weights α and β, and η 1 and η 2 are a coordinate system defined by the transformation ( ) 1+ξ1 η 1 = 2 1, η 2 = ξ 2. 1 ξ 2 This transformation effectively maps the triangular master element into a quadrilateral one as shown in Figure 3.4. The η-coordinate system, as viewed within the ξ-coordinate system, can be interpreted as a so-called collapsed coordinate system. It can be noted that there is a singularity in this transformation at ξ 2 = 1; however, this singularity does not appear in the final form of the basis functions. With the definitions of the principal functions ψ (1) i and ψ (2) ij, the triangular basis functions are now constructed as their tensor product φ ij (ξ 1,ξ 2 ) = ψ (1) i (η 1 ) ψ (2) ij (η 2). (3.2) 20

35 η 2 ( 1, 1) ξ 2 ( ) 1+ξ1 η 1 = ξ 2 ( 1, 1) (1, 1) ( 1, 1) ˆΩ ξ 1 (1, 1) η 2 = ξ 2 ( 1, 1) ˆΩ η 1 (1, 1) η = 1 η = 0 η = 1 Figure 3.4: Mapping of Triangular Master Element to Quadrilateral The orthogonality of this basis over the domain of the master triangular element can easily be demonstrated using the orthogonal properties of the Jacobi polynomials P (α,β) n that are used to construct the principal functions ψ i and ψ ij of the basis. This orthogonality property, of course, leads to a diagonal mass matrix, which can be trivially inverted. Specifically, the orthogonality relation can be shown as ˆΩ φ ij φkl dâ = = 2 (2i+1)(i+j +1) if i = k and j = l 0 otherwise. (3.3) These can be ordered hierarchically, e.g., constant function, linear functions, quadratic functions, etc., as follows φ 1 = φ 00 φ 2 = φ 01 φ 3 = φ 10 φ 4 = φ 02 φ 5 = φ 11 φ 6 = φ 20 φ (k+1)(k+2)/2 k = φ 0k φ (k+1)(k+2)/2 = φ k0 (3.4) A p =1 or linear element is obtained by using the first three functions, a p = 2 or quadratic element by using the first six, and so on. The first ten basis functions as defined by equation 3.2 are shown in figure

36 Figure 3.5: Triangular Hierarchical Basis Functions Rectangular Elements Master Rectangular Element Similar to the triangular case, a reference or master rectangular element ˆΩ is defined using a local Cartesian coordinate system, (η 1,η 2 ) [ 1,1] as defined in Figure 3.6. The vertices of the master element are numbered in a counter-clockwise manner, and edge number i is defined as in figure

37 η 2 Edge 1 ( 1, 1) 4 3 (1, 1) Edge 2 ( 1, 1) ˆΩ 1 2 Edge 3 Edge 4 η 1 (1, 1) Figure 3.6: Master Rectangular Element A physical element in the (x,y) coordinates can be defined as the image of this master element under the affine transformation T e : ˆΩ Ωe defined by x = 1 [ (1 η1 ) x1 + ( ] ) 1+η 1 x2 2 y = 1 [ (1 η2 ) y1 + ( ] ) 1+η 2 y4 2 where (x 1,y 1 ), (x 2,y 2 ), (x 3,y 3 ), and (x 4,y 4 ) are the physical coordinates of the vertices of Ω e numbered locally in a counter-clockwise manner. The Jacobian of this transformation is J = (x, y) (η 1, η 2 ) = 1 [ x2 x 1, 0 2 0, y 4 y 1 ] J = A( ) e 4, where A ( ) e is the area of Ω e. The inverse of the transformation T 1 e : Ω e ˆΩ is given by η 1 (x) = 1 ( ) 2x x1 x 2 x η 2 (y) = 1 ( ) 2y y1 y 4 y 23

38 R : (x, y) (η 1, η 2 ) (x 4, y 4 ) (x 3, y 3 ) η 2 ( 1, 1) (1, 1) Ω e y (x 1, y 1 ) (x 2, y 2 ) ( 1, 1) ˆΩ η 1 (1, 1) x Figure 3.7: Master Rectangular Element Transformation where x = x 2 x 1 and y = y 4 y 1. It should be noted that the Jacobian for a general quadrilateral element is not constant, e.g., it is a function of η 1 and η 2. This means the value of the Jacobian would need to be computed and stored at each quadrature point when evaluating the integrals, which leads to an extremely large data storage requirement. A non-constant Jacobian also increases the order of quadrature rule that needs to be used to obtain exact integration over the master element. For these two unfavorable reasons, the choice in this work is limited to rectangular elements, where the Jacobian remains constant. To combat the errors that would result where non-rectangular shapes are needed, e.g., to fit complicated boundaries, triangular elements are used, hence the concept of a mixed element mesh. 24

39 Given these transformations the area integrals over the physical element in (x, y) coordinates can be related to the area integrals over the master element in local coordinates (η 1,η 2 ). Specifically the relationship is given by ( ) e f da = A Ω e 4 f dâ, ˆΩ where it can be seen that due to the choice of rectangular elements, instead of general quadrilaterals, the constant Jacobian can be pulled out of the integral. Rectangular Basis An orthogonal basis for the rectangle can be constructed as a tensor product of the so-called principal functions defined as ψ (1) i = P (0,0) i (η 1 ), ψ (2) j = P (0,0) j (η 2 ). wherep (α,β) n isthen-thorderjacobipolynomialofweightsαandβ, whereinthiscase α = β = 0, which results in the Legendre polynomials, i.e., the Legendre polynomials are a special subset of the Jacobi polynomials. With the definitions of the principal functions ψ (1) i and ψ (2) j, the rectangular basis functions are now constructed as their tensor product φ ij (η 1,η 2 ) = ψ (1) i (η 1 ) ψ (2) j (η 2 ). (3.5) As mentioned earlier, given the fact that DG methods do not require continuity of the approximate solution between elements, quadrilateral element basis functions may be developed that exclude the usual cross-terms that are present in standard 25

40 C 0 elements, e.g., a linear quadrilateral element may be used instead of a bilinear quadrilateral element. The orthogonality of this basis over the domain of the master rectangular element can easily be demonstrated using the orthogonal properties of the Legendre polynomials P (0,0) n that are used to construct the principal functions ψ i and ψ j of the basis. This orthogonality property, again, leads to a diagonal mass matrix, which can be trivially inverted. Specifically, the orthogonality relation can be shown as ˆΩ φ ij φkl dâ = = 4 (2i+1)(2j +1) if i = k and j = l 0 otherwise. (3.6) As in the case of the triangle, these also can be ordered hierarchically, e.g., constant function, linear functions, quadratic functions, etc., as follows φ 1 = φ 00 φ 2 = φ 01 φ 3 = φ 10 φ 4 = φ 02 φ 5 = φ 11 φ 6 = φ 20 φ (k+1)(k+2)/2 k = φ 0k φ (k+1)(k+2)/2 = φ k0. (3.7) The first ten basis functions as defined by equation 3.5 are shown in figure D Elements In going from two dimensions to three dimensions many of the elemental properties remain the same in (x,y) with just the addition of a parameter or function in the vertical direction, e.g., the master element transformations, the inverse transformations, basis functions, etc. With regard to domain discretization, the vertical parameter can lead to some difficulties in using DG methods for free surface flow. A key feature of the developed DG methods is the discretization of all the primary variables using discontinuous polynomial spaces of arbitrary order, including the free 26

41 Figure 3.8: Rectangular Hierarchical Basis Functions surface elevation. In a standard Cartesian-coordinate system, this results in elements in the surface layer having mismatched lateral faces, i.e., a staircase boundary as seen in figure 3.9, which requires the use of some form of mesh smoothing [6]. This difficulty will be avoided by employing a so-called sigma-coordinate system in the vertical, which transforms both the free surface and bottom boundaries into coordinate surfaces. The top sigma-coordinate surface, which corresponds to the free surface, will be discretized using two-dimensional meshes consisting of triangular and/or quadrilateral elements, as seen in section 3.1. This two dimensional mesh is then extended in the vertical direction to produce a three-dimensional mesh of one or more layers, N, of triangular prismatic and hexahedral elements. The mapping from Cartesian to sigma-coordinates is shown in Figure With the sigma-coordinate 27

42 z = ζ (+) (x, y,t) h z = 0 z = ζ ( ) (x, y,t) h h(x, y) Figure 3.9: Staircase boundary introduced by the use of a standard Cartesian coordinate system. transformation in place, it can be seen in the subsections to follow that the elemental implementation in three-dimensions is rather straightforward Triangular Prism Element Master Triangular Prism Element A reference or master triangular prism domain ˆΩ is defined using a local Cartesian coordinate system, (ξ 1,ξ 2,ξ 3 ) [ 1,1] with ξ 1 +ξ 2 0, i.e., half of the unit cube, as defined in Figure

43 z z = ζ( x,ỹ) x z = 0 σ = 0 z = h( x,ỹ) σ = 1 x = x, y = ỹ, σ = z ζ h+ζ, t = t σ x σ = 0 σ = 1 Figure 3.10: Sigma-Coordinate Transformation 29

44 ξ 3 ( 1,1,1) 3 ξ 2 2 (1, 1,1) 1 ( 1, 1, 1) ξ 1 ( 1,1, 1) 6 5 (1, 1, 1) 4 ( 1, 1, 1) Figure 3.11: Master Triangular Prism Element A physical element in the (x,y,σ) coordinates can be defined as the image of this master element under the affine transformation T e : Ω e ˆΩ defined by x = 1 2 y = 1 2 [ (ξ1 +ξ 2 ) x1 ( 1+ξ 1 ) x2 ( 1+ξ 2 ) x3 ] [ (ξ1 +ξ 2 ) y1 ( 1+ξ 1 ) y2 ( 1+ξ 2 ) y3 ] σ = 1 ] [1 2n+ξ 3 2N where (x 1,y 1 ), (x 2,y 2 ), and (x 3,y 3 ) are the physical coordinates of the vertices of Ω e on the top face numbered locally in a counter-clockwise manner, N is the number of σ-layers, and n refers to the n th σ-layer in Ω e. 30

45 The Jacobian of this transformation is J = (x, y, σ) (ξ 1, ξ 2, ξ 3 ) = 1 2 x 2 x 1, x 3 x 1, 0 y 2 y 1, y 3 y 1, 0 0, 0, 1/N J = A( ) e 4N = V e ( ) 4 where V ( ) e is the volume of Ω e, which is the product of the area A e = 1 2( x1 y 2 +x 2 y 3 + x 3 y 1 x 1 y 3 x 2 y 1 x 3 y 2 ) of the top (or bottom) face of Ωe and the thickness, 1/N, of each σ-layer. The inverse of the transformation T 1 e : ˆΩ Ωe is given by ξ 1 = 1 A e { (y3 y 1 ) [ x 1 2 ( x2 +x 3 ) ] + ( x 1 x 3 ) [ y 1 2 ( y2 +y 3 ) ]} ξ 2 = 1 A e { (y1 y 2 ) [ x 1 2 ( x2 +x 3 ) ] + ( x 2 x 1 ) [ y 1 2 ( y2 +y 3 ) ]} (3.8) ξ 3 = 2Nσ +2n 1. With these transformations in place, integrations performed over the volume of Ω e can be computed over ˆΩ via ( ) e f dv = V Ω e 4 For area integrations over the faces of Ω e, we have ˆΩ e f dˆv. ( ) e f da = A f dâ l Ω e 4 }{{ lˆω } lateral (side) faces, ( ) e f da = A f dâ xyω e 2 }{{ xyˆω } top and bottom faces. 31

46 Triangular Prism Basis An orthogonal basis for the triangular prism can be constructed as a product of the set of 2D triangular basis functions and Legendre polynomials in the vertical. In terms of the so-called principal functions they are defined as ψ (1) i = P (0,0) i (η 1 ), ψ (2) ij = where P (α,β) n ( 1 η2 2 ) i P (2i+1,0) j (η 2 ), ψ (3) k = P (0,0) k (η 3 ). is the n-th order Jacobi polynomial of weights α and β, and η 1, η 2, η 3 are a coordinate system transformation defined over a suitably defined reference element which can be seen in figure With the definitions of the principal functions ψ (1) i, ψ (2) ij, and ψ(3) k the triangular prism basis functions are now constructed as their tensor product φ ijk (ξ 1,ξ 2,ξ 3 ) = ψ (1) i (η 1 ) ψ (2) ij (η 2) ψ (3) k (η 3), which can be written more suitably as a function of two-dimensional basis functions as { } φ 3D k = { } φ 2D P (0,0) k (η 3 ). The use of these basis functions results in several desirable properties in terms of numerical implementation. Like in 2D, these basis function are orthogonal over the domain of the master triangular prism element, which leads to a diagonal mass matrix, 32

47 η 3 η 2 η 1 ξ 1 = (1+η1)(1 η2) 2 1 ξ 2 = η 2, ξ 3 = η 3 ξ 3 ) η 1 = 1+ξ1 2( 1 ξ 2 1 η 2 = ξ 2, η 3 = ξ 3 ξ 2 ξ 1 Figure 3.12: Mapping of Triangular Prism Element to Hexahedral 33

48 or what is known as a matrix free implementation. Specifically, the orthogonality relation can be shown as ˆΩ φ ijk φlmn dˆv = = 8 (2i+1)(2i+2j +2)(2k +1) if ijk = lmn 0 otherwise. (3.9) The basis functions are also hierarchical as in the 2D case Rectangular Hexahedron Elements Master Cubic Element Areferenceormastercubicelement ˆΩisdefinedusingalocalCartesiancoordinate system, (η 1,η 2,η 3 ) [ 1,1] as defined in Figure 3.13 with the vertices of the master element numbered in a counter-clockwise manner. A physical element in the (x,y,σ) coordinates can be defined as the image of this master element under the affine transformation T e : ˆΩ Ωe defined by x = 1 2 y = 1 2 [ (1 η1 ) x1 + ( 1+η 1 ) x2 ] [ (1 η2 ) y1 + ( 1+η 2 ) y4 ] σ = 1 ] [1 2n+η 3 2N where(x 1,y 1 ),(x 2,y 2 ),(x 3,y 3 ),and(x 4,y 4 )arethephysicalcoordinatesofthevertices of Ω e numbered locally in a counter-clockwise manner, N is the number of σ-layers, and n refers to the n th σ-layer in Ω e. 34

49 ( 1,1,1) 4 η 3 3 (1,1,1) η 2 2 (1, 1,1) 1 ( 1, 1,1) η 1 8 (1,1, 1) ( 1,1, 1) 5 6 ( 1, 1, 1) 7 (1, 1, 1) Figure 3.13: Master Cubic Element The Jacobian of this transformation is J = (x, y, σ) (η 1, η 2, η 3 ) = 1 2 x 2 x 1, 0, 0 0, y 4 y 1, 0 0, 0, 1/N J = V e ( ) 8 where V ( ) e is the volume of Ω e, which is the product of the area A e of the top (or bottom) face of Ω e and the thickness, 1/N, of each σ-layer. The inverse of the transformation T 1 e : Ω e ˆΩ is given by η 1 (x) = 1 ( ) 2x x1 x 2 x η 2 (y) = 1 ( ) 2y y1 y 4 y η 3 (σ) = 2Nσ +2n 1 35

50 where x = x 2 x 1 and y = y 4 y 1. With these transformations in place, integrations performed over the volume of Ω e can be computed over ˆΩ via Cubic Basis ( ) e f dv = V Ω e 8 ˆΩ e f dˆv. An orthogonal basis for the cube can also be constructed as a product of the set of 2D basis functions, in this case the rectangular set, with the Legendre polynomials in the vertical. In terms of the so-called principal functions they are defined as ψ (1) i = P (0,0) i (η 1 ), ψ (2) j = P (0,0) i (η 2 ), ψ (3) k = P (0,0) k (η 3 ). where P (α,β) n is the n-th order Jacobi polynomial of weights α and β, which in the case of the cube α = β = 0 for all principal functions resulting in the special case of Legendre polynomials. With the definitions of the principal functions ψ (1) i, ψ (2) j, and ψ (3) k the triangular prism basis functions are now constructed as their tensor product φ ijk (η 1,η 2,η 3 ) = ψ (1) i (η 1 ) ψ (2) j (η 2 ) ψ (3) k (η 3). which can be written more suitably as a function of two-dimensional basis functions as { φ 3D } k = { φ 2D } P (0,0) k (η 3 ). 36

51 As in the prismatic case, the use of these basis functions results in several desirable properties in terms of numerical implementation. Like in 2D, these basis function are orthogonal over the domain of the master cubic element which leads to a diagonal mass matrix. Specifically, the orthogonality relation can be shown as ˆΩ φ ijk φlmn dˆv = = 8 (2i+1)(2j +1)(2k +1) if ijk = lmn 0 otherwise. (3.10) The basis functions are also hierarchical as in the 2D case. 3.3 Element Basis Function Subroutine The Fortran subroutine developed for this work, orthogonal basis.f, evaluates the orthogonal basis functions (as described in sections 3.1.1, 3.1.2, 3.2.1, 3.2.2) and their derivatives at a given point, i.e., a quadrature point, for a 2D triangle, a 3D triangular prism, a 2D quadrilateral, or a 3D hexahedral. The subroutine takes the following inputs: DEFINITION VARIABLE NAME INPUT CHOICES Element Type: ELEM Triangle or Quadrilateral Dimension: DIM 2 or 3 Polynomial Degree of Basis: D Positive Integer Coordinate Point: PT Size DIM and outputs two matrices, BASIS and DBASIS, of size (D+2)(D+1)(DIM 1) 2 1 and (D+2)(D+1)(DIM 1) 2 DIM, respectively, containing the basis functions and the gradients of the basis functions evaluated at the coordinate point, respectively. 37

52 The defined basis functions and their derivatives can be constructed using a combination of standard recurrence relations and identities for Jacobi polynomials; see, for example, [1]. These relations can then be used to numerically evaluate the basis functions at a particular point, e.g., (ξ 1,ξ 2 ), within the element needed for computation. However, in the case of the triangle and triangular prism, even though the basis functions are polynomials in terms of the ξ coordinates, the recurrence relations will have a coordinate singularity, making their numerical evaluation difficult. This difficulty can be avoided by employing the singularity-free recurrence relations developed in [20], which work directly in the original ξ coordinates of the triangular element rather than the collapsed η coordinates that are used in the approach that works directly with the Jacobi polynomials. Singularity-free recurrence relations for the prism elements can be obtained by using the relations developed in [20] for the triangular elements in (ξ 1,ξ 2 ) along with standard Legendre polynomial recurrence relations in ξ 3. After the basis functions and their derivatives are calculated by the recurrence relations, they are then reordered hierarchically before being output into the BASIS and DBASIS matrices. Pseudo code of the subroutine can be seen in figure

53 SUBROUTINE ORTHOGONAL BASIS. Algorithm for returning basis functions and gradients of basis functions evaluated at a coordinate point for triangle, triangular prisms, quadrilaterals, and hexahedrals. IF Element equals triangle or quadrilateral IF Coordinate point is contained within element DO Appropriate recurrence relation in each variable ENDDO IF Dimension equals 2 DO Reorder hierarchally and output matrix ENDDO ELSEIF Dimension equals 3 DO Apply recurrence relation in third dimension, then reorder hierarchally and output matrix ENDIF ELSE Output error message END SUBROUTINE ORTHOGONAL BASIS Figure 3.14: Pseudo Code of Orthogonal basis.f 39

54 CHAPTER 4 NUMERICAL QUADRATURE AND CUBATURE As the development of computational methods and algorithms continues to evolve, the necessity of performing multi dimensional integration over a variety of domains appears more commonly. FEM is rapidly progressing in multi dimensional and multi regional domains with the use of triangle, square, tetrahedra, hexahedra and triangular prism elements. As noted in the previous chapters, a crucial point in computing these FEM solutions is the evaluation of the domain integrals arising over the master element in the discrete local weak form of the governing PDEs, which in two dimensions take the form and in three dimensions I [ f ] 2D I [ f ] 3D = f(ξ,η) dâ (4.1) ˆΩ f(ξ,η,ζ) d V. (4.2) ˆΩ Numerical integration formulas, more commonly known as numerical quadrature (in 1D or 2D) or cubature (in 3D) rules, are frequently used in computational mathematics to approximate the definite integral of a function f over a region ˆΩ via a weighted sum of function evaluations, i.e., f dˆω ˆΩ N w i f(x i ) (4.3) i=1 40

55 wherew i arethequadratureorcubatureweightsandx i arethequadratureorcubature points at which the function f is evaluated. In the case of integrating polynomials, the approximately equals,, in the formula above can be replaced by an equality, =, i.e., the numerical integration can be made exact, if a sufficient number of weights and points are used. A numerical integration formula is said to be of degree p if it integrates all polynomials of degree p exactly. In one dimension, numerical quadrature rules are well developed and understood, where an N-point Gauss Legendre rule is exact for all polynomials of orders up to and including 2N 1. In one-dimension, the Gauss Legendre rules are the optimal rules, meaning the number of points used is the theoretical lower bound to integrate a polynomial exactly. However, no general theory regarding the optimal number of quadrature points in two and three dimensions is known. In fact, the situation in multiple dimensions is considerably more complex. While the interval is the only connected compact subset of R 1, regions of R 2 and R 3 come in an infinite variety of shapes, each with its own topological features. These complexities lead to the necessity that each shape will require a different set of numerical integration formula and will have its own theories regarding optimality. Following [26], p degree integration formulas in more than one dimension can be classified into two broad categories: product and nonproduct formulas. For nice regions in n dimensions (e.g., the n simplex, the n cube, various pyramids and prisms, etc) product formulas can be constructed using combinations, or products, of formulas developed for regions of dimension < n. These types of rules include, for example, double and triple product rules for the square and the cube, respectively, which use one dimensional Gauss formulas in each direction. A similar approach can also be 41

56 used for the triangle, the tetrahedron, and the triangular prism after using appropriate Duffy type transformations [8] to map the shapes to the n cube. While the use of product rules is straightforward and easily allows for the construction of integration formula of degree p in multiple dimensions, the resulting rules are inefficient in that they use a far greater number of function evaluation points than necessary to evaluate the integral of a polynomial of degree p exactly, especially as n and p increase. Shape specific nonproduct formulas, on the other hand, offer an efficient alternative to the use of product rules for calculating integrals in multiple dimensions. These types of rules can be developed using, for example, the general method of polynomial moment fitting as described in detail in section 4.3. For example, in two dimensions, efficient nonproduct rules have been developed for both the triangle [9, 17, 25] and the square [5, 11, 27], and in three dimensions, efficient nonproduct rules have been developed for the tetrahedron [14, 18, 25, 30, 26] and the cube [4, 10, 12, 19, 26]. A lower bound for the number of points N required for an even p = 2k degree rule for a region in R n is given by [26]; specifically, Theorem 1 If p = 2k is even, then a formula of degree p contains at least N = (n+k)! n!k! (4.4) points where n is the dimension. Now consider p odd, p = 2k +1. It is clear that Theorem 1 implies N odd N even. (4.5) Other than this implication there has been no other generalization for when p is odd for general shapes in R n. 42

57 Inanefforttogainefficiencyinthetwo andthree dimensionaldgfemcodes,an extensive literature search was conducted to identify the best available shape-specific, nonproduct rules for the triangle, square, and the cube, i.e., rules with a number of points N as close to the lower bounds presented above. The results of this search are presented in the tables in the following sections, which compare the number of quadrature/cubature points N needed for p degree product and nonproduct rules for the triangle, square, triangular prism, and cube. In each case, the nonproduct rules result in fewer points than the product approaches (with exceptions occurring when the rules result in the same formula for all approaches) with many of the formulas resulting in optimal or close to optimal rules. The efficiency is a representation of the percent more efficient the nonproduct rule is over the product. In the course of this literature search, it was discovered that no such nonproduct rules existed for the triangular prism domain. Thus, in an effort to gain efficiency in the three dimensional DG FEM code when using triangular prism elements, a set of nonproduct rules specifically for the triangular prism have been developed here. Again, to the author s knowledge, there have been no such nonproduct rules developed specifically for the triangular prism. These types of rules are derived and presented in detail in Section Triangle Quadrature Rules The efficient nonproduct rules utilized in the code for area integrations over triangular elements have been obtained from an extensive search of the quadrature literature; see, for example, [9, 17, 23, 25]. These nonproduct rules range in efficiency from 22% 39% compared to the use of double product rules, which again are the 43

58 Polynomial Double Triangular Degree Product Non-Product Efficiency Quality p Rule Rule 1 1 1* [9] PI 2 4 3* [9] 25.00% PI 3 4 4* [17] PI 4 9 6* [30] 33.33% PI 5 9 7* [9] 22.22% PI [9] 25.00% PI * [13] 25.00% PI [9] 36.00% PI [9] 24.00% PI [30] 30.56% PI [30] 22.22% PI [30] 32.65% PI Table 4.1: Number of N points for various rules over the triangle. Denotes optimal lower bound. successive application of one dimensional Gauss rules after applying a Duffy transformation that maps the triangle to a square (see Chapter 3). The triangle is one of the most common shapes in finite element meshes, which has lead to many studies searching for quadrature rules, both numerically and analytically. As a result, very high order rules exist, up to p = 30, though only those up to p = 25 are considered for the finite element work presented here. All triangular rules which are made use of are so-called PI quality P indicating that all weights are positive and I indicating that all points are inside the reference domain. A summary of the number of points up to degree 12 is presented in Table 4.1 and even degree rules are shown in Figure 4.1, where the symmetry of the rules, which simplifies their derivation, can be noted. 44

59 (a) p = 2 (b) p = 4 (c) p = 6 (d) p = 8 (e) p = 10 (f) p = 12 Figure 4.1: Quadrature rules for the triangle. 45

60 4.2 Square Quadrature Rules The implementation of square elements in the DG FEM code was a foundational part of the work developed here. A comprehensive search of square quadrature literature was conducted to obtain the most efficient, while still maintaining PI quality, nonproduct rules; see, for example, [5, 11, 27]. The quality standard was achieved for all rules except p = 23, in which the only nonproduct rule that currently exists is NI N indicating one or more of the weights are negative. Though not PI, this rule is still worthy of consideration over a double product rule due to the facts that only one weight is negative, all the points are still inside the domain, and it yields a 22.22% efficiency savings. The overall range of efficiency savings is between 22% 39% compared to the use of double product Gauss rules. Square elements are less commonly used in FEM than triangular elements, however a substantial amount of research exists on the development of their quadrature rules. While not as complete as triangles, there still have been high order rules developed (up to p = 31), although there are quite a few missing degrees in between. For the scope of this work, rules up to p = 23 were included. A summary of the number of points up to degree 12 is presented in Table 4.2 and even degree rules are shown in Figure 4.2, where the symmetry of the rules, which simplifies their derivation, can be noted. 4.3 Triangular Prism Cubature Rules As noted previously, prior to this work no such nonproduct rules have been developed specifically for the triangular prism. These types of rules are derived and presented in detail in this section. 46

61 (a) p = 2 (b) p = 4 (c) p = 6 (d) p = 8 (e) p = 10 (f) p = 12 Figure 4.2: Quadrature rules for the square. 47

62 Polynomial Double Square Degree Product Non-Product Efficiency Quality p Rule Rule 1 1 1* [26] PI [26] 25.00% PI 3 4 4* [26] PI 4 9 6* [29] 33.33% PI 5 9 7* [26] 22.22% PI * [29] 37.50% PI * [15] 25.00% PI [29] 36.00% PI * [24] 32.00% PI [27] 38.89% PI * [24] 33.33% PI [27] 36.74% PI Table 4.2: Number of N points for various rules over the square. Denotes optimal lower bound. Table (4.3) compares the number of cubature points N needed for p degree rules for the triangular prism for p = 1 through p = 5 for two types of product rules, the nonproduct rules for the triangular prism developed here, and the lower bound for a formula of degree p (given by Theorem 1). The triple product rules listed use the above mentioned approach of transforming the triangular prism to the cube and then applying one dimensional Gauss formulas in each of the three directions. This approach is the least efficient method for numerically integrating over the triangular prism. The double product approach listed uses a combination of the best available nonproduct rules from the literature for triangles in conjunction with one dimensional Gauss rules in the remaining direction. While this approach is more efficient than using the triple product formulas, it still results in a greater number of evaluation points than required. Finally, these two types of product rules are compared to the 48

63 Polynomial Triple Double Triangular Prism Degree Product Product Non-Product Efficiency Quality p Rule Rule Rule * [21] PI * [21] 33.33% PI [21] 16.67% PI * [21] 25.00% NI [21] PI [21] 38.89% PI [21] 19.04% PI Table 4.3: Number of N points for various rules over the triangular prism. Denotes optimal lower bound number of cubature points N required for the nonproduct rules developed here. In each case, the nonproduct rules result in fewer points than the triple and double product approaches (with the exception of p = 1, which results in the same formula for all three approaches) with some of the formulas resulting in optimal or close to optimal rules. On average, the triple product rules require nearly twice as many function evaluation points and weights as the new triangular prism rules, and the best available double product rules require, on average, nearly one and a half times as many points and weights. The development of the newly created rules is organized in the next section as follows: In section 4.3.1, the method of construction by use of polynomial moment fitting and the newly created symmetry groups used in the derivation of the cubature rules are presented; section presents an example derivation of a low order cubature rule using the method of polynomial moment fitting; and section contains the triangular prism cubature formulae developed for polynomial degrees 1 through 5 as well as figures of the cubature rules. 49

64 ζ ( 1,1,1) η (1, 1,1) ( 1, 1,1) ξ ( 1,1, 1) (1, 1, 1) ( 1, 1, 1) Figure 4.3: Reference Triangular Prism Method of Construction To begin, a reference triangular prism domain Ω is defined using a Cartesian coordinate system (ξ,η,ζ) [ 1,1], ξ +η 0, i.e., half of the unit cube, as defined in Chapter 3 and as shown again in Figure 4.3. Weconsiderthevolumeintegralofacompletepolynomialf = f(ξ, η, ζ)ofdegree p over this domain, that is, I [ f ] f(ξ,η,ζ) dξ dη dζ 3D Ω where f(ξ,η,ζ) = a ijk ξ i η j ζ k for i+j +k p, (4.6) i,j,k 50

65 with the polynomial coefficients a ijk being arbitrary and the number of terms being M = (p+1)(p+2)(p+3). 6 Calculating the integral of this polynomial over the prism gives us the so-called moments used in the derivation of the cubature formula. For example, the first few moments would be I [ f ] [ = a 1 [4 ]+a 2 4 [ ]+a 3 4 [ ] [ ] 4 ]+a 4 0 +a , (4.7) where the arbitrary coefficients a have been conveniently redefined using a single index. Now a cubature rule of degree p computes the integral of all polynomials of degree p exactly via a weighted sum of functions evaluations at N points, that is, I [ f ] N = w i f(ξ i,η i,ζ i ) i N N N = a 1 w i + a 2 w i ζ i + a 3 w i ζ i i i i N N + a 4 w i η i + a 5 w i ξi (4.8) i i Given that the coefficients a i are arbitrary, the following set of M nonlinear equations, which are obtained by setting a given moment of equation (4.7) to the corrersponding sum of equation (4.8), must be satisified for the cubature rule to be 51

66 exact a 1 : N w i = 4, a 2 : i=1 N w i ξ i = 4 3 i=1 a 3 : N w i η i = 4 3, a 4 : i=1 N w i ζ i = 0 i=1 N a 5 : w i ξi 2 = 4 3,. i=1 These are the so-called moment equations, which are functions of the unknown cubature points, (ξ i,η i,ζ i ), and their respective weights, w i. This general procedure for deriving cubature rules is known as the method of polynomial moment fitting. It has been successfully used to derive nonproduct rules for a variety of two and three dimensional shapes Symmetry groups Enforcing some level of symmetry on the location of the cubature points greatly simplifies the task of solving the system of M nonlinear moment equations. For this reason, we are interested in deriving symmetric rules for the triangular prism. One exception occurs for the optimal rule found for p = 2, which is a nonsymmetric rule that was derived using a procedure described in [26] for optimal second degree formulas for arbitrary regions of dimension n. We construct six symmetry groups over the triangular prism. Each cubature point in a given group carries the same weight w i. The first group, n 1, consists of just one point, the centroid of the reference triangular prism. Obviously, there can be at most one of these per cubature rule. Group n 2 has two distinct points at ±ζ where ζ 0 along the vertical centerline of the reference prism (i.e., ξ = η = 1/3). Group n 3 has three symmetric points all lying on the medians of the triangle in the plane ζ = 0. 52

67 Group n 4 is similar to group n 3 except it has two sets of points at ±ζ where ζ 0, giving it six distinct points. Group n 5 includes six symmetric points in the ζ = 0 plane, none of which are on the medians. Lastly, group n 6 is the same as group n 5 except it has two sets of points at ±ζ where ζ 0, giving it twelve distinct points. The symmetry groups are illustrated in Figure 4.4 and a summary of the groups of points, weights, and number of unknowns per group is shown in Table 4.4. n 1 n 2 ζ = 0 +ζ n 3 n 4 ζ n 5 n 6 Figure 4.4: Symmetry groups used for the triangular prism Due to the points lying on the median in symmetry groups n 3 and n 4, ξ and η can be parameterized as a function of one variable, say a, and in a similar way groups n 5 and n 6 can be parameterized as a function of two variables, say b and c. The outcome 53

68 is a significantly reduced set of unknowns which therefore leads to simplifications in solving the system of equations. Symmetry Number of Number of Unknowns Group Points per Unknowns per per n j Group Group Group n w n w,ζ n w,a n w,a,ζ n w,b,c n w,b,c,ζ Table 4.4: Distribution of points and unknowns for symmetry groups. The use of the above symmetry groups leads to another very important simplification. A rule which is symmetrical about the centerline will integrate all terms, despite their order, which are antisymmetrical about that axis. In others words, equations need not to be considered which correspond to terms in which the exponent of ζ i is odd. It should be pointed out that these equations should not be discarded, but are just not considered due to the symmetry requirements. The advantageous result is a reduced number of independent simultaneous nonlinear moment equations for which a solution must be found. This becomes particularly important as the degree p increases due to the simplification of the number of M equations it provides. The permutation of groups which results in the number of points, N, resulting from a given rule is often referred to as the structure of the cubature rule. Let K i be the number of permutations of symmetry group of type j, then the number of points 54

69 produced by a rule can be determined by, N = K 1 +2K 2 +3K 3 +6K 4 +6K 5 +12K 6 (4.9) And the number of n unknowns is, n = K 1 +2K 2 +2K 3 +3K 4 +3K 5 +4K 6 (4.10) With the simplification that symmetry and structure provide all that remains is to determinethecombinationofgroupsn 1 throughn 6 withthesmallestnumberofpoints for which a solution can be obtained for the system. There are several problematic outcomes that can occur when solving the remaining system of nonlinear equations 1. The system may be found to be inconsistent (i.e. no solution) 2. The solution may consist of imaginary numbers If neither of these instances result, then the solution of points and weights is categorized as PI, PO, NI, or NO (O indicates some points are outside the reference domain). In our derivations we strive to find optimal PI rules Solution of System of Equations In this subsection, an example of the derivation of a low order cubature rule for the triangular prism is presented. We begin by constructing a complete polynomial function f of degree p = 2 in three dimensions using the definition given by equation (4.6), f(ξ,η,ζ) = [1 ξ η ζ ξ 2 ξη η 2 ξζ ηζ ζ 2 ]{a} The volume integral of f over the triangular prism domain is 55

70 1 ζ= 1 1 η= 1 η ξ= 1 f(ξ,η,ζ)dξdηdζ = [4]a 1 [ 4 3 ]a 2 [ 4 3 ]a 3 +[0]a 4 +[ 4 3 ]a 5 +[0]a 6 +[ 4 3 ]a 7 +[0]a 8 +[0]a 9 +[ 4 3 ]a 10 (4.11) The integration of f by a cubature rule having N points is, N w i f(ξ i,η i,ζ i ) = [w 1 f(ξ 1,η 1,ζ 1 )+...+w i f(ξ i,η i,ζ i )+...+w N f(ξ N,η N,ζ N )]{a} i 1 N N N = a 1 w i +a 2 w i ξ i +...+a 10 w i ζi 2 (4.12) i=1 i=1 i=1 Using the definition given in (4.3) and equating the right hand side of (4.11) and (4.12) and also noting that the coefficients a i are arbitrary leads to the following system of M = 10 nonlinear equations, a 1 : a 2 : a 3 : a 4 : a 5 : N w i = 4 a 6 : i=1 N w i ξ i = 4 a 7 : 3 i=1 N w i η i = 4 a 8 : 3 i=1 N w i ζ i = 0 a 9 : i=1 N w i ξi 2 = 4 3 i=1 a 10 : N w i ξ i η i = 0 i=1 N w i ηi 2 = 4 3 i=1 N w i ξ i ζ i = 0 i=1 N w i η i ζ i = 0 i=1 N w i ζi 2 = 4 3 i=1 56

71 Based on the simplification that the symmetry of the ζ coordinate provides, the system of equations simplifies to a 1 : a 2 : a 3 : a 5 : N w i = 4 a 6 : i=1 N w i ξ i = 4 a 7 : 3 i=1 N w i η i = 4 3 a 10 : i=1 N w i ξi 2 = 4 3 i=1 N w i ξ i η i = 0 i=1 N w i ηi 2 = 4 3 i=1 N w i ζi 2 = 4 3 i=1 The optimal lower bound of N points is given by (4.9) p = 2 k = 1 N = (3+1)! 3!1! The next step is to find a combination of groups n 1 through n 6 that gives N = 4 points and a solution exists. We begin by trying 1 n 1 group and 1 n 3 group, which by using equations (4.9) and (4.10) leads to = 4. N = 1(1)+3(1) = 4 points n = 1(1)+2(1) = 3 unknowns;w 1,w 2,a. By inspection it can immediately be seen that this permutation will give an inconsistent set of equations, that is, a 10 : N w i ζi 2 = i=1 We move on to the only other combination which could lead to a symmetric solution with 4 points which consists of 2 n 2 groups. Using equations (4.9) and (4.10) leads 57

72 to, N = 2(2) = 4 points n = 2(2) = 4 unknowns;w 1,w 2,ζ 1,ζ 2 Equations a 1,a 2,and a 3 yield, a 1 : w 1 +w 2 = 2 a 2 : w 1 +w 2 = 2 a 3 : w 1 +w 2 = 2 Since a 1,a 2,and a 3 are linearly dependant, only one equation needs to be retained, we will retain a 1. Now solving equation a 5 yields, a 5 : w 1 +w 2 = 6 It can nowbe seen that this combination also leads to an inconsistent set of equations, i.e., w 1 +w 2 = 2 w 1 +w 2 = 6 Since a symmetric solution for the optimal number of points can not be found, we now try to find a combination of groups n 1 through n 6 that gives N = 5 points and a solution exists. Before we proceed it should be noted that a nonsymmetrical optimal PI solution was obtained for the triangular prism using the method outlined in [26]. The results of this cubature rule are presented later. 58

73 The first permutation for 5 points we will try is 1 n 2 group and 1 n 3 group, which by using equations (4.9) and (4.10) leads to, N = 2(1)+3(1) = 5 points n = 2(1)+2(1) = 5 unknowns;w 1,w 2,ζ 1,a This grouping generates the following system of equations, a 1 : 2w 1 +3w 2 = 4 a 2 : w 1 ( 2 3 ) w 2 = 4 3 a 3 : w 1 ( 2 3 ) w 2 = 4 3 a 5 : w 1 ( 2 9 ) w 2(6a 2 +4a+1) = 4 3 a 6 : w 1 ( 2 9 ) w 2(3a 2 +2a) = 0 a 7 : w 1 ( 2 9 ) w 2(6a 2 +4a+1) = 4 3 a 10 : 2w 1 (ζ 2 ) = 4 3 We only need to retain equations which are independent from one and other, which from this system are a 1,a 5,a 6, and a 10. The solution of m independent nonlinear equations requires that at least as many n unknowns are provided, given m n. This leaves a nonlinear system of m = 4 equations and n = 4 unknowns, so we can proceed onward to finding a solution. Solving the system results in, ±ζ = 3 2 a( 4 27a 2 +18a ) ( 4 27a 2 +18a )1 2 w 1 = 18a2 +12a 9a 2 +6a+1 4 w 2 = 27a 2 +18a+3 Where a turns out to be a free parameter giving the system infinitely many solutions. Based on the parameterization done a [-1,0], so choosing any value within a s 59

74 p = 1, N = 1, error = 0, quality, = PI Weight ξ η ζ N otes: OPTIMAL Table 4.5: Triangular Prism Cubature Rule: p = 1. domain will result in a symmetric PI cubature rule. In the results presented here we choose a to be -1, which gives the following 5 point PI symmetric cubature rule for p = 2, w 1 = 3 2 ζ = ± 2 3 w 2 = 1 3 a = Some Cubature Formulas for Triangular Prisms Presented in Tables are the cubature rules derived over the triangular prism for degrees p = 1 through p = 5 where the specific rules can been seen in Figure 4.5. The notation used in the tables is as follows: p = Highest degree polynomial for which quadrature rule is precise N = Number of quadrature points and weights error = The largest relative error between exact and numerical integration quality = Quality of the cubature rule (i.e. PI, PO, NI or NO) 4.4 Cube Cubature Rules The novel implementation of square elements in two-dimensions was extended in three-dimensions to the cube for use in the DG FEM code. A wide-ranging search 60

75 p = 2, N = 4, error = e 15, quality = PI Weight ξ η ζ N otes: OPTIMAL Antisymmetric Table 4.6: Triangular Prism Cubature Rule: p = 2 a. p = 2, N = 5, error = 0, quality = PI Weight ξ η ζ Table 4.7: Triangular Prism Cubature Rule: p = 2 b. p = 3, N = 6, error = 0, quality = NI Weight ξ η ζ N otes: OPTIMAL Table 4.8: Triangular Prism Cubature Rule: p = 3 a. 61

76 p = 3, N = 8, error = e 14, quality = PI Weight ξ η ζ Table 4.9: Triangular Prism Cubature Rule: p = 3 b. p = 4, N = 11, error = e 13, quality = PI Weight ξ η ζ Table 4.10: Triangular Prism Cubature Rule: p = 4. 62

77 (a) p = 1 (b) p = 2 unsymmetric (c) p = 2 symmetric (d) p = 3 (e) p = 4 (f) p = 5 Figure 4.5: Quadrature rules for the triangular prism. 63

78 p = 5, N = 17, error = e 13, quality = PI Weight ξ η ζ Table 4.11: Triangular Prism Cubature Rule: p = 5. of cube quadrature literature was carried out to obtain the most efficient nonproduct rules; see, for example, [4, 10, 12, 19, 26]. A best attempt has been made to find rules that have positive weights with all points located inside the cube, however for some of the higher degrees there have been no such rules developed, consequently some have negative weights and/or points outside the unit cube. In choosing which rule to use, PI rules took precedence over rules which had a lower number of points which were not PI. Rules in which a point lay outside the unit cube were only considered if the point was slightly outside, i.e., rule p = 12, ξ = , which is only slightly outside of the domain. The overall range of efficiency savings is between 5% 40% compared to the use of triple and double product rules. A summary of the number of points up to degree 12 is presented in Table 4.12 and even degree 64

79 Polynomial Triple Double Cube Degree Product Product Non-Product Efficiency Quality p Rule Rule Rule * [10] PI [10] PI * [10] 25.00% PI [10] 22.22% PI [10] 33.33% PI [19] 5.00% PI [19] 20.83% PI [12] 27.50% PI [12] 27.50% PI [12] 31.82% PI [12] 37.50% PI [10] 30.41% NO Table 4.12: Number of N points for various rules over the cube. Denotes optimal lower bound. rules are shown in Figure 4.6, where the symmetry of the rules, which simplifies their derivation, can be noted. 4.5 Quadrature Subroutine The Fortran subroutine developed for this work, quadrature.f, takes a user input of p, the polynomial degree of the rule, and region, and returns the individual points and weights of the p degree quadrature rule for the given region as well as the total number of points, N. Regions are denoted as edge, triangle, square, triangular prism (triprism), or cube. Regions, the domain on which they are defined, and the highest degree of polynomial for which a quadrature rule exists in the subroutine is as follows: 65

80 (a) p = 2 (b) p = 4 (c) p = 6 (d) p = 8 (e) p = 10 (f) p = 12 Figure 4.6: Quadrature rules for the cube 66