An efficient linear elastic FEM solver using automatic local grid refinement and accuracy control
Highlights
► Efficient Finite Element multigrid solver with an accuracy control strategy. ► Automatic local grid refinement. ► Linear time complexity and minimal memory requirement. ► Examples up to 200 million degrees of freedom are presented.
Introduction
Industrial problems lead to larger and larger finite element models reaching several hundred million degrees of freedom (dof). This paper presents some developments concerning an efficient finite element algorithm and its implementation.
The main features are:
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a controlled accuracy level which is imposed a priori (before the calculation). Over-accurate (hence too costly) or under-accurate (hence possibly unsafe) solutions are avoided.
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a fast solver and a quasi-minimal problem size. Very large problem sizes are targeted.
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an automatic 3D mesher. The mesh is automatically built and reflects the mechanical problem. The meshing step is faster, safer and requires less expertise.
To achieve these objectives, an algorithm based on several discretisation levels is used, leading to a locally adaptative multigrid solver.
During the process of numerically analysing a physical problem, three main error sources are encountered: a modeling error (not discussed here), a discretisation error and a numerical error (which has to be reduced as fast as possible to be smaller than the discretisation error).
The discretisation error appears when a continuous problem is reduced to a set of discrete equations. The discretisation error depends on how the mesh adapts itself to the mechanical problem (element size, type, geometry, boundary conditions, loads, etc.). The design of an optimal mesh has been a particular point of interest for several decades: see Thompson et al. [1], [2], Shephard [3], Babuska et al. [4], [5], [6], [7], Southwell [8], [9].
To build an optimal mesh, hence ensuring a certain level of accuracy, error estimators are used. Under the generic term of error estimators, different techniques to estimate the discretisation error are grouped (including error indicators). The dynamic mesh construction based on global or local error estimators is the basic concept of adaptive techniques.
One of our goals is to obtain a quasi-uniform precision over the entire domain. To achieve this accuracy, several strategies are available, for example:
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the h-method: element subdivision.
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the p-method: interpolation function enrichment.
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the hp-method: a combination of the two previous cases.
In this work, an h-method is used with a local error indicator; to obtain a targeted accuracy at a minimum cost, the mesh size has to vary over the domain. Several ways to build an adapted mesh have been used. Initially, when a solution was considered as locally inaccurate, the geometry was entirely remeshed to improve the solution quality in the concerned areas. Only conforming elements were used and element sizes were diminished continuously from coarse regions to fine regions; for example see Kornhuber and Krause [10], Aubry et al. [11] and Bornemann et al. [12]. In this work, a more recent and local non-conforming meshing technique is used.
Numerical simulations in various fields (fluid mechanics, solid mechanics, electromagnetism, etc.) lead to large systems of equations. Since the 1970s, Brandt [13], [14], [15] has developed multilevel methods to address the solution of such large systems. Venner and Lubrecht [16] applied and developed these techniques for contact mechanics. Adams [17] proposed an adaptation of these techniques to finite element computations. Multigrid techniques are often presented from the sole point of view of efficient solvers. However, some cited references explicitly propose error estimators.
One of the major contributions of this paper is to clearly demonstrate the efficiency reached with the combination of two multigrid technique advantages: an efficient convergence of numerical errors and a simple and intrinsic error estimation based on the comparison of solutions obtained on different discretisation levels. The multigrid solver can be seen as an iterative multiscale strategy in order to automatically capture different scales (structure, high gradient zones, local non-linearities, etc.) which is coupled with an error estimator in order to locally refine to reach a given level of accuracy, see Becker et al. [18] and Bastian and Wittum [19].
The local mesh refinement algorithm uses non-conforming meshes. The entire problem does not need to be remeshed at every step of the refinement strategy to incorporate a layer of ‘transition’ elements as in Wieners [20] and Schmidt and Siebert [21]. In the proposed strategy, hanging nodes (also referred later as non-coinciding nodes) are used to define the fine problem, see Jones and Jimack [22] or Janssen and Kanschat [23]. Becker and Braack [24] study two algorithms with conforming and non-conforming meshes; non-conforming techniques are found to be more efficient. Those two techniques are also developed for finite differences, for example see Wise et al. [25] where adaptive multigrid techniques using non-conforming nodes are used for a strongly anisotropic physical problem. Brown and Lowe [26] studied adaptive multigrid techniques for gravitational wave simulations and especially interpolation/restriction order influence on convergence speed.
Cavin et al. [27], [28] developed these methods for linear elastic finite element calculations (static and dynamic). Biotteau et al. [29], [30], [31] or Ekevid et al. [32] integrated non-linear material properties. Although Cavin et al. or Biotteau et al. demonstrated the potential of the method with respect to the accuracy control, the programming environment (Cast3M, [33], [34]) was not efficient enough to completely reveal the algorithm performance in terms of computational cost. The interlevel communication, data structure, etc. were not suited to this multilevel purpose. Indeed, multigrid techniques make intensive use of several discretisation levels and an efficient communication between these different levels is required. The performance gain is conditioned by a limited computing cost of this interlevel communication. This is especially the case for the interpolation/restriction operators. The implementation has to be made at a sufficiently low level to ensure that both data structures and operator implementations are very efficient. Basically, a linear complexity evaluation and a linear memory requirement have been ensured.
A dedicated multilevel finite element implementation has been developed from scratch to demonstrate the performance of the proposed method. This paper describes the main features of the code and describes some obtained results.
Section snippets
Theoretical framework
Very classical static linear elastic isotropic homogeneous and small displacement assumptions are used, for instance see Hughes [35]. The discrete problem is reduced to a set of linear equations where K is a stiffness matrix, u is the nodal displacements and f is the applied load:
The algorithm presented in this paper merges an accuracy control, an efficient multilevel solver and an automatic 3D mesher. To construct an automatic mesh refinement which ensures the prescribed accuracy, three
Efficiency
The method efficiency is based on three distinct aspects:
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An efficient solver.
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A localised problem, thus a quasi-minimal problem size.
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A dedicated implementation.
Results
Multigrid solver for global refinement: Some 2D results are presented up to 200 million dof using 1 CPU (2.0 GHz), 110 GBytes of RAM and less than 4 h of computing time. Below, some 3D calculations on more complex parts are shown. These 3D calculations used up to 167 million dof using 1 CPU and 228 GBytes of RAM in less than 42 h.
Localisation: The huge gain in memory requirement and computation time is illustrated. For the maximum accuracy tested, the developed code requires 1 day and 80 GBytes of
Conclusion
An efficient FEM solver based on a multigrid strategy has been developed. A local refinement technique is used to dynamically construct a mesh which ensures a prescribed accuracy. An accuracy indicator evaluation is cheap and is naturally derived from the refinement strategy using comparisons between different discretisation levels. The implementation reconciles contradictory objectives: memory requirement and computational cost (complexity). Several numerical examples covering a very large
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