April 23, Abstract

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1 The FCM compared to the h-version FEM for elasto-plastic problems Alireza Abedian 1, Jamshid Parvizian 2, Alexander Düster 3, Ernst Rank 4 (1. Department of Mechanical Engineering, Daneshpajoohan Higher Education Institute, Isfahan, Iran 2. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran 3. Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik (M-10), Technische Universität Hamburg-Harburg, Hamburg, Germany 4. Lehrstuhl für Computation in Engineering, Fakultät für Bauingenieur- und Vermessungswesen, Technische Universität München, München, Germany) April 23, 2014 Abstract Abstract The Finite Cell Method (FCM) combines the high-order finite element method and the fictitious domain approach for the purpose of simple meshing. In the present study, the FCM is applied to the Prandtl-Reuss flow theory of plasticity and the results are compared with an h-version Finite Element Method (h-fem). The numerical results show that the FCM is more efficient compared to h-fem for elasto-plastic problems, although the mesh does not conform to the boundary. It is also demonstrated that the FCM performs well for elasto-plastic loading and unloading. Key words octree. 1 Introduction Finite Cell Method, h-version Finite Element Method, Prandtl-Reuss, quadtree, The Finite Cell Method (FCM) [1, 2] can be regarded as an embedding or fictitious domain method [3 5], combined with the p-version finite element method [6]. The efficiency of the p-version Finite Element Method (p-fem) for problems of elasto-plasticity has been thoroughly investigated [7 12]; the p-version yields clearly superior accuracy in comparison with its h-version counterparts. Whilst the FCM has been fairly well investigated as an efficient method for linear problems, little has been done to explore its merits in elasto-plastic structural analysis [13 15]. In the FCM, the polynomial degree of the finite element approximation is increased (p-version) to improve the accuracy of the solution, rather than following the traditional mesh refinement approach (h-version). Furthermore, different local refinement strategies have been developed [16,17] for different types of discontinuities and singularities to improve the quality of the FCM approximation. The aim of this paper is to numerically compare the accuracy of the FCM for the Prandtl-Reuss flow theory of plasticity with an h-version FEM. The computations are based on the elasto-plastic J 2 flow theory, using the von Mises yield criterion. Further assumptions include small displacements and small strains. The results show that the FCM makes the analysis of complex geometry problems very convenient and easy as well as reliable. The accuracy of stresses for nearly incompressible materials is always of interest. Incompressibility effects may be introduced by the elasto-plastic material law and may slow down the convergence. The numerical examples Received??.??,?? / Revised??.??,?? Corresponding author Alireza Abedian, Assistant Professor, Ph. D., abedian.a@gmail.com 1

2 presented in this paper demonstrate that a good convergence of stress components can be obtained using FCM, even if the point of interest is located within the plastic region. The paper is organized as follows: Section 2 explains the FCM for elasto-plastic formulation and a modified quadtree integration scheme. Section 3 presents numerical examples used to evaluate the efficiency of the FCM compared to h-fem, for the J 2 plasticity. This is followed by a summary and a conclusion. 2 The Finite Cell Method 2.1 Elasto-plastic formulation The finite cell method is a combination of high-order FEM and fictitious domain methods; we therefore use the well-established terminology of the FEM in this paper. To solve the weak form of the equilibrium equation using the high-order finite cell method, the domain of interest Ω, as shown in Fig. 1, is extended into a domain Ω e, i.e., Ω Ω e, only with the advantage that Ω e can be discretized with a Cartesian structured grid of cells. Ω e \ Ω Ω + = Ω e Figure 1: The domain Ω is extended to Ω e [1]. The weak formulation of the equilibrium conditions of a solid body reads as finding the displacement field u which satisfies the (homogeneous) Dirichlet boundary conditions, so Ω e α ε(v) : σ(u)dω = α v b dω+ v tdγ (1) Ω e Γ N in which b is the vector of volume loads, t is the vector of prescribed tractions on the Neumann boundary Γ N, the operator : denotes the double-contracted product and v is the test function. α is defined as α = { 1.0 in Ω, 0.0 otherwise. (2) In each cell, the displacement field u is approximated using hierarchical shape functions based on integrated Legendre polynomials [18]. Hierarchic bases have been used to implement two different types of Ansatz spaces in two dimensions [6, 19]: the trunk space S p ξ,p η ts (Ω q st ) and the tensor product space S p ξ,p η ps (Ω q st ). Three different types of Ansatz spaces have been implemented in three dimensions [6, 19]: the trunk space S p ξ,p η,p ζ ts (Ω h st ), the tensor product space S p ξ,p η,p ζ ps (Ω h st) and the space S p,p,q (Ω h st). The physically nonlinear problem under contemplation is the J 2 flow theory for small strains with nonlinear isotropic hardening [20]. The strains are small and can be decomposed into an elastic and a plastic component. The stresses should satisfy both the constitutive equations and the equilibrium conditions. Since the nonlinear stress(σ)- strain(ε) relationship depends on the load history, the weak formulation is a nonlinear functional to be solved incrementally. Using the Newton-Raphson method for each step [t (n),t (n+1) ] to linearize the weak formulation, it is 2

3 possible to compute a sequence of equilibrated load steps: Ω e α ε(v) : C (i) (n+1) : ε( u(i+1) )dω = u (i+1) (n+1) α v b (n+1) dω+ v t (n+1) dγ Ω e Γ N α ε(v) : σ (i) (u)dω (3) Ω (n+1) e = u (i) (n+1) + u(i+1) C (i) (n+1) := σ(i) (n+1) ε (i) (n+1) ( ) (i) (n+1) denotes a variable at the ith Newton-Raphson iteration during the load step in[t (n),t (n+1) ]. The actual stress state σ (i) as well as the algorithmic tangent modulus C(i) have to be recomputed at each step i [20]. In linear (n+1) (n+1) elasticity problems the stress(σ)-strain(ε) relationship is linear. The tangent modulus tensor in Equation (3) denotes the elasticity tensor and the equation is solved in one step. The spatial discretization of Equation (3) is performed using the FCM. 2.2 Integration scheme In embedded FE methods, the boundary of the physical domain is not captured by the mesh. Rather, accurate integration schemes may capture the boundary. One of such schemes used for FCM is a modified quadtree integration scheme. Quadtree is a tree data structure used for partitioning a domain by recursively subdividing it into four quadrants. The extension of a quadtree for 3D space is called octree, in which each internal node or domain has eight children [21]. With the help of a quadtree or an octree, it is easy to perform the appropriate numerical integration for Equation (3) using the composed integration over subcells (Fig. 2). η ξ η ξ s r y x Parent cell R = 5 Local coordinate system Figure 2: Composed integration following the relationship of (x, y) and (r, s) for 5 levels of quadtree refinement (R). A mapping has to be applied to establish a relationship between the coordinates of the subcell (r,s) and the cell (ξ, η) [2]. In this investigation, we used a modification of the standard quadtree integration scheme which has two specific features (Fig. 3): (I) Since the subcells are much smaller than the cell, the function over the subcells could be regarded as a lowerorder function; it is accordingly possible to apply low-order Gauss quadrature. In the standard p-fem, p + 1 Gauss points are used in each direction for the elements of order p [8]. We therefore begin with p + 1 Gauss points in each direction; then we reduce the number of Gauss points incrementally down to 1 following the refining of the quadtree. Any integration point outside the boundary of the physical domain defined by the quadtree is ignored. Fig. 3(a) shows the position of active Gauss points of subcells mapped from the local coordinate (r,s) to the parent cell. 3

4 (II) The second feature involves avoiding the ill-conditioning inherent in fictitious methods by assigning a small α to the Gauss points outside the physical domain [1]. Therefore, the cell is reconsidered with p+1 Gauss points in each direction and the points that are outside the physical domain are only taken into account to avoid ill-conditioning of the problem (Fig. 3(b)). As an example, 144 integration points are used for integrating over a cell of order p=4 with 3 levels of quadtree refinement (R = 3), as shown in Fig. 3. η ξ η ξ (a) (b) Figure 3: An adaptive integration scheme consists of a set of integration points inside (a) and outside (b) the physical domain. 3 Numerical examples In each of the examples, the units of dimensions and displacements are in [mm], and stresses in [MPa]. The material is assumed to be elastic/perfectly-plastic with a shear modulus of µ = [MPa], bulk modulus κ = [MPa] and yield stress σ yield = 450 [MPa], [22]. The plane strain situation and a unit thickness are assumed for 2D problems. The computations for FCM are conducted using the p-fem code AdhoC [23], and the direct solver SPOOLES [24] is used to solve the overall equation system. The results are visualised by Paraview [25] using a fine postprocessing mesh. In this paper we will restrict our investigations to a quadtree/octree refinement level of R = 3, and the trunk space S p ξ,p η ts (Ω q st ), S p ξ,p η,p ζ ts (Ω h st ). The FCM leaves the mesh unchanged and progressively increases the degree p, of the piecewise polynomial approximation, until some desired level of precision is reached. In the standard h-fem, however, the mesh is refined and the polynomial degree of the shape functions remains unchanged. The h-version mesh generation and computations were carried out with the commercial finite element package ABAQUS [26] which employs special pressure-displacement elements for incompressible materials, based on a mixed finite element formulation. These elements also include an additional, internal mode trial function [27]. Since it is difficult to create a structured mesh when faced with complex geometries, we made use of the unstructured mesh generating scheme. We chose the 2D and 3D enhanced low-order approximation elements CPE4R (p = 1) and C3D8R (p = 1) in h-fem. The reduced integration scheme and a dense mesh are used to avoid locking and hour-glassing phenomena [27]. 3.1 Perforated plate The first numerical example to be considered is a square plate with a central perforation under cyclic loading. A quarter of the square plate is modelled and the symmetry conditions are imposed on the lower and right sides of the plate (Fig 4). The uniform traction t n = 100 is scaled with a factor 4.5 λ 4.5 in 205 load steps. As a reference solution we use the results provided by Wieners [22] based on a very fine mesh with Q2P1 elements. Results of interest are the stress σ yy at point a=(9,0), the stress σ xx at point b=(6.1953,3.8047), the displacements u x at point c=(0,10) and u y at point d=(10,10). 4

5 The discretization of h-fem and the p-version of the FCM are adjusted to achieve almost the same degrees of freedom (Dofs) for comparison. Fig. 5 shows 100 cells used with p = 10, which corresponds to 9600 Dofs, and 4686 elements of h-fem, which corresponds to 9622 Dofs. t y = λ t n c d λ t n 300 h=10 y x w=10 b a r = 1 load (MPa) load step Figure 4: Perforated square plate under plane strain condition with cyclic loading. Figure 5: Perforated square plate discretized by 100 cells (left) and 4686 elements (right). 5

6 Reference FCM h-fem σyy at point a load factor λ Figure 6: The stress component σ yy at point a Reference FCM h-fem σxx at point b load factor λ Figure 7: The stress component σ xx at point b. 6

7 Reference FCM h-fem ux at point c load factor λ Figure 8: The displacement u x at point c Reference FCM h-fem 0.01 uy at point d load factor λ Figure 9: The displacement u y at point d. As shown in Figs. 6-9, the displacements are approximated accurately by both methods, but the stresses are more accurate using the FCM. Let us consider this problem again with a load factor of λ = 4.5 and study convergence behaviour in terms of Dofs. The first and last mesh used for h-fem are shown in Fig. 10. For FCM calculations, the polynomial degree varies from 4 to 14 using 100 cells (Fig. 5). The results are plotted in the form of relative errors in a double logarithmic scale in Figs. 11 and 12. As these figures indicate, the FCM yields smoother, more accurate results particularly at point a, where the most critical stress concentration occurs. Using FCM, only 4740 Dofs with a corresponding polynomial degree p = 7 are needed to compute the stress components with a relative error less than 7

8 5%. Considering these numerical investigations, it is apparent obvious that it is easy to obtain a degree of accuracy suitable for practical implementation by means of the FCM. Figure 10: Perforated square plate discretized by 777 and 9770 elements. 10 FCM h-fem Relative error [%] 1 5 % relative error 0.1 Dofs Figure 11: Relative error ( σ yy,re f σ yy,fcm σ yy,re f 100[%]) at point a, load factor λ =

9 FCM h-fem Relative error [%] 10 5 % relative error 1 Dofs Figure 12: Relative error ( σ xx,re f σ xx,fcm σ xx,re f 100[%]) at point b, load factor λ = Perforated thick plate For the second example, we consider a thick, perforated plate with a unit thickness under uniform traction of t n = 100 that is raised monotonously to a factor of λ = 4 in 49 load steps (Fig. 13). Symmetry boundary conditions are applied at y = 0,z = 0 and x = 10 planes. By way of a reference solution, we use the p-version FEM with 199 hexahedral elements of p = 10 (Fig. 14), whereas the curved boundary has been represented precisely with blending functions [6]. The result of interest is the stress σ yy at the critical point e= (9,0,0), where the stress concentration occurs. 9

10 t y = λ t n t z y x 400 w λ t n e r h load load step Figure 13: Thick perforated plate under a uniform load. Figure 14: Thick perforated plate discretized by 199 hexahedral elements. 100 cells with p=8 are used for the FCM calculations, which in this case corresponds to Dofs. For the h-version calculations, elements give Dofs (Fig. 15). 10

11 Figure 15: Thick perforated plate discretized using 100 cells and elements. Considering the results in Fig. 16, the FCM approach for p=8 conforms better to the reference solution than the h-fem. It is evident that the FCM provides a more accurate approximation using higher order p. This shows that, the most accurate results can be achieved using FCM σyy at point e Reference FCM h-fem load factor λ Figure 16: The stress component σ yy at point e. A three-dimensional view of the von Mises stress for the FCM with p=8 and the p-version reference solution at the last load step λ = 4 are shown in Fig

12 Figure 17: The von Mises stress results using the FCM (left) and p-version FEM (right) for the last load step λ = 4. Since two different codes, ABAQUS and AdhoC, are used in the present study, it is difficult to achieve a fair comparison for the CPU computational time. It is, however, safe to conclude that it is faster to pursue solution convergence in the FCM, because it has a fixed mesh and the polynomial order is only increased for the sake of a more accurate approximate solution. It is also possible to use a fixed dense pattern of integration points for a p- extension, allowing the entire pre-processing procedure including generating the Cartesian mesh and setting up the integration points to be performed in the beginning of the computation. Besides, the FCM enjoys all the advantages of the p-version FEM, such as condensing interior degrees of freedom which are confined to the element. The elimination of interior degrees of freedom can be performed in parallel on element level without any communication. This increases the efficiency of the overall parallel approach which takes advantage of the fact, that the computation of element matrices of high order constitutes a significant part of the overall numerical effort. Furthermore it can be shown that the elimination of the interior degrees of freedom improves also the conditioning of the resulting global equation system which is important when using iterative solvers [28 31]. 4 Conclusion This paper compared the FCM numerically with an h-version FEM for the Prandtl-Reuss flow theory of plasticity. The results demonstrate the efficiency of the FCM to solve materially nonlinear problems. Numerical examples show that the FCM provides efficient and accurate approximations to this class of physically non-linear problems. It is easier to control the discretization error with the FCM than with the h-fem, so, the degree of error can be reduced by increasing the order of elements and not necessarily by employing a new mesh. Error monitoring and control is achieved automatically, with minimum user interaction and without substantial loss of computational efficiency. For two- and three-dimensional benchmark problems, we find that the models based on the FCM deliver much more accurate results compared with their h-version counterparts. To summarise, the application of the FCM in elastoplastic structural analysis can be recommended. These observations suggest that it would be beneficial to extend the use of the FCM to problems of elastoplasticity with finite strains. Results in this context will be presented in forthcoming papers. 12

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