Efficient hp-finite elements

Transcription

1 Efficient hp-finite elements Ammon Washburn July 31, 2015 Abstract Ways to make an hp-finite element method efficient are presented. Standard FEMs and hp-fems with their various strengths and weaknesses are compared. Strategies to allow the hp-fems to be faster are described and examples are shown. 1 Introduction The finite element method is heavily used in industry, especially in solid and fluid mechanics. It allows the simulation of complex two and three dimensional problems using very simple piece-wise polynomials. The normal approach is to use small degree polynomials and a large number of elements. The elements quickly grow in these settings and the problem becomes intractable. In order to get accurate results for a small number of elements, there has been a need to develop hp FEMs which allow for exponential rates of convergence [2]. Melenk et al and Ainsworth et al seek to develop hp FEMs that have the same asymptotic computation count as the standard finite element methods. They have both developed a basis of polynomials with the needed properties to allow for efficient evaluation of their integrals. Melenk et al developed a basis for hexahedral elements [2] while Ainsworth et al developed a basis for simplicial elements [1]. I would like to verify their results and compare the two ways to see if they are compatible. I will first provide a background of FEMs to illustrate the importance of hp FEMs. I will explain the two methods which allow for easy evaluation of the integrals in hp-fe. Then I will explain what Melenk et al and Ainsworth et al did and describe further directions to improve these methods. 2 Background Boris Galerkin was a Polish mathematician who developed ways to solve problems dealing with infinite spaces. He developed a way to find an approximate answer to an infinite dimensional problem. The problem is this: Find u V satisfying a(u, v) = f(v) v V where f is a functional and a is bilinear. Galerkin posed a similar but easier problem: Consider a finite-dimensional sub-space V n of the infinite dimensional Hilbert space V which we hope is dense in V or close enough. The solution for this problem is easy: In this sub-space V n it is enough to find u n so that a(u, e i ) = f(e i ) for every basis element e i V n In terms of FE methods, the finite dimensional space is piece-wise polynomials of degree n. The 1

2 Figure 1: The approximate solution from the finite dimensional space might or might not be close enough to the real solution. Depends on your finite dimensional space. differential equation is turned into it s weak formulation and all polynomials are iterated in the basis through this weak formulation. Each element gives a local stiffness matrix and the matrices are assembled into a global stiffness matrix which is then solved to find the coefficients on the basis functions to make up the approximate solution. In order to make the approximation better, hitherto it was necessary to make lots of elements with a low degree or to keep a small number of elements but with the polynomial degree very high. The hp-fem seeks to combine the advantages of both while still having the same optimal operation count that these first methods had. 3 Sum Factorization Typically for calculating two dimensions, a double summation has this form: q 0 q 1 s=1 t=1 Φ 1 i (x s )Φ 2 j(y t )Ψ 1 k(x s )Ψ 2 l (y t )w s w t b(x s, y t ) Where Φ and Ψ come from the basis functions, w s and w t are weights associated with a quadrature formula, b(x s, y t ) is a map from a master element to the element currently being worked on, and i, j, k and l all index the one dimensional basis functions. This requires n 4 q 0 q 1 number of operations! However, if some pieces are factored out of the first summation and summed up first, then it only requires n 2 q 1 (n 2 + q 0 ) operations. q 0 s=1 Φ 1 i (x s )Ψ 1 k(x s ) q 1 t=1 Φ 2 j(y t )Ψ 2 l (y t )w s w t b(x s, y t ) This is a very simple operation, but it requires the basis to have some sort of tensor product, or in the case of simplicial elements, some transformation to a tensor polynomial [1]. 4 Spectral Methods Eventually it will become necessary to use some quadrature rule to evaluate the integrals. Because the basis functions are piece-wise polynomials, Gauss- 2

3 Figure 2: There are usually associated nine degrees of freedom with a square element Lobatto quadrature rules can exactly evaluate the integrals. Knowing this, the shape polynomials (the polynomials that are zero on the boundary but give the element shape in the middle) will be designed to be zero at most of the points of the quadrature rule. This will drastically decrease the number of evaluations. Consider again q 0 s=1 Φ 1 i (x s )Ψ 1 k(x s ) q 1 t=1 Φ 2 j(y t )Ψ 2 l (y t )w s w t b(x s, y t ) What if most of Φ 1 i (x s)φ 2 j (y t)ψ 1 k (x s)ψ 2 l (y t) was zero? Then the operation count would be n 2 q 1(n 2 +q 0) where q 1, q 0 << q 1, q 0. As an example, it requires about fifty function evaluations to exactly approximate the integral of a hundred degree polynomial but with classical spectral methods, it would require one point. However, over integration is important to insure that the method is stable. This is done by writing the basis functions of degree p with Lagrange interpolants l i based on the p + q + 1 points of Gaussian quadrature. l i (ξ j ) = 0 for all but q + 1 points where ξ j is the j th node in Gaussian quadrature of p + q + 1 points. This over integration allows you to pick just how many extra evaluations (q) that you want to add. Note that usually p is in the range of a hundred while q is in the range of ten. 5 Hp-spectral Galerkin FEM Melenk et al provide a basis and method for hp-fem called the hp-spectral Galerkin FEM that can take advantage of both sum factorization and the spectral method. For simplicity of presentation, I will describe the 2D form of the basis functions, though the authors believe a similar extension to 3D is easy enough to see because of the tensor nature of the basis functions. 5.1 Basis functions In figure 2, there are four black dots, a 1, a 2, a 3, a 4, on the vertex nodes that represent the simple bilinear nodes. The function space E 0 of these functions is defined by the following, h 1 (x) = 1 x, h 2 (x) = x, E 0 = h i (x) h j (y) (i, j) 1, 2 1, 2 3

4 When these functions are bilinear, they are a lot easier to deal with. That is why standard FEM uses low degree polynomials. However, spectral FEM uses high degree polynomials in the interior to increase convergence to piece-wise analytic solutions. The interior function space J takes care of the degree of freedom represented by a 9 and is made up as follows: l i (x, GL p+q ) = p+1 i=1 i j x x i, J = l i (x, GL p+q ) l j (y, GL p+q ) 2 i p, 2 j p x j x i Note that although there are p + q + 1 points in GL p+q only p + 1 points are used to make the Lagrange interpolants and the boundary points are always used to make sure they are zero on all the sides. To accurately represent the side nodes, there are many different options that the authors give to write the shape functions. The side shape functions are functions that are zero on every side except one side and take care of the degrees of freedom of the four red dots on the sides, a 5, a 6, a 7, a 8. I will give an example of how to write the function spaces E 1, E 2, E 3, E 4 of side shape functions that also take advantage of the spectral method. Note the h i and l i (x, GL p+q ) are same as before. E 1 = l i (x, GL p+q ) h 1 (y) 2 i p E 2 = E 3 = E 4 = l i (x, GL p+q ) h 2 (y) 2 i p h 1 (x) l i (y, GL p+q ) 2 i p h 2 (x) l i (y, GL p+q ) 2 i p In this example, the same degree is used as the interior functions, but it is flexible enough to handle smaller degrees than the interior. However, the functions will no longer be using the same quadrature rule. Each degree of freedom and the functions associated with them could have their own quadrature rule to which they were adapted. This might be useful if the problem had some inherent scale that you could take advantage of, like modeling a wave front. 5.2 Comparison of asymptotic work By using sum factorization and spectral methods with these basis functions, the order of complexity is reduced from O(p 6 ) to O(p 4 ) in two dimensions. (See table 1 for a comparison in three dimensions.) In this case the p represents the polynomial power of the side shape functions as those are the functions less suited to the quadrature rule. If lower degree polynomials were used instead like in the standard FEM but the high degree interior functions were used as well then hp-spectral FEM has the same asymptotic work as the standard FEM. Of course the method is superior in problems where the solution is piecewise analytic. It requires very few elements to be computed because of the high degree of the interior polynomial. 4

5 Version of hp FEM 2-D 3-D Standard O(p 4 (p + q) 2 ) O(p 6 (p + q) 3 Sum factorization O(p 4 (p + q) + p 2 (p + q) 2 ) O(p 6 (p + q) + p 4 (p + q) 2 + p 2 (p + q) 3 ) Hp-spectral O(p 4 (1 + q) + p 2 q 2 ) O(qp 6 + p 5 + q 2 p 4 + q 3 p 2 ) Table 1: This table is from Melenk et al [2] 6 Bernstein-Bézier FEM In a hexahedral element, the tensor product of the polynomials is natural with finite elements because control of the values at the boundaries is crucial. However, in simplicial elements the tensor product structure doesn t allow for that. Ainsworth et al [1] develop a basis that allows for the control of the boundaries but still allows the use of sum factorization as well. 6.1 Bernstein Polynomials Given a non-degenerate set of corners in the simplex x 1,..., x d+1, define the barycentric coordinates λ = (λ 1,..., λ d+1 ) of any point x R d uniquely by d+1 d+1 x = λ k x k, 1 = k=1 k=1 Define the Bernstein polynomials as ( ) n Bα n = λ α, α Id n α where the standard multi-index notation is used throughout and Id n = α Z d+1 + α = n. I will list some standard multi-index notation for the reader. λ α = d k=1 λα k k, α = d k=1 α k, ( n d α) = n!/α!, α! = k=1 α k!. These Bernstein polynomials have desirable properties. They form a basis for the piece-wise polynomial space, they can match and be continuous across the elements by just looking at the boundary points on the elements, the product of two Bernstein polynomials is just a scaled Bernstein polynomial, and there is a closed form solution for the evaluation of the polynomials. However, in the form the Bernstein polynomials are currently in, we can not take advantage of sum factorization. 6.2 Duffy transformation The Duffy transformation allows the conversion of the Bernstein polynomials into a tensor product construction. The Duffy transformation takes a point t on the unit square in d dimensions and maps it to a point x on a triangle in d dimensions. To get the coordinates you use the following equations: λ 1 = t 1, λ 2 = t 2 (1 t 1 ), λ 3 = t 3 (1 t 1 )(1 t 2 ) λ k 5

6 . λ d = t d (1 t 1 )(1 t 2 )...(1 t d 1 ), λ d+1 = (1 t 1 )(1 t 2 )...(1 t d ) Looking where the corners of the n-dimensional cube get mapped to will illuminate what the transformation is doing. In a n-dimensional cube there are 2 d corners which get mapped to the d + 1 corners of the simplicial element. 2 d i cube corners will get mapped to λ i or the i th corner of the simplicial element for 1 i d and the only point mapping to λ d+1 is (0, 0,..., 0) which accounts for all the 2 d points in a d-dimensional cube. Effectively, the corners of a cube are squeezing together to make a tetrahedral element. Applying this change of coordinates to the Bernstein polynomials, they become a product of one dimensional Bernstein polynomials. Bα(x(t)) n = Bα n 1 (t 1 )Bα n α1 2 (t 2 )...B n α1... α d 1 α d (t d ) Now just integrate over the d-dimensional cube and that will integrate over the tetrahedral element. In these new coordinates the Bernstein polynomials have the tensor product structure needed for sum factorization. Note that with this arrangement that the spectral element methods discussed before can t be used but the Bernstein polynomials are zero on the sides for most of the α Id n which decreases the operation count a little. 7 Further Research The Bernstein-Bézier elements have very desirable properties that make for very efficient evaluation of the elements but the last property that would allow for highly efficient calculations would be an added spectral property that would allow only a few non-zero evaluations to calculate the integrals. This could be done using a different basis for the interior that preserved some of the properties of the Bernstein polynomials but allowed for spectral methods. For the hp-spectral Galerkin elements there needs to be a more rigorous analysis of the stability and convergence of the methods used to calculate the integral, especially since the boundary functions and the interior functions aren t the same degree of polynomial. Numerical studies as well as analysis would help to see when the added complexity of implementing these methods significantly added to the convergence of the method with fewer operations than standard FEM. 8 Conclusion The hp-spectral Galerkin and Bernstein-Bézier FEMs provide for faster convergence to the actual solution with the optimal asymptotic number of computations as normal FEMs. Hp-spectral Galerkin FEMs use high degree tensor product Lagrange interpolants in the interior to obtain faster convergence and then uses sum factorization and spectral methods to decrease the operation count. Bernstein-Bézier FEMs use the properties of Bernstein polynomials and the Duffy transformation to be able to use sum factorization to decrease the 6

7 operation count. Bernstein polynomials have a number of other properties that allow for efficient evaluation such as analytic moments and derivatives. Further areas include more rigorous analysis on the stability and convergence of these methods as well as numerical tests. References [1] Mark Ainsworth, Gaelle Andriamaro, and Oleg Davydov. Bernstein-bézier finite elements of arbitrary order and optimal assembly procedures. SIAM Journal on Scientific Computing, 33(6): , [2] Jens M Melenk, Klaus Gerdes, and Christoph Schwab. Fully discrete hpfinite elements: Fast quadrature. Computer Methods in Applied Mechanics and Engineering, 190(32): ,