Adaptive FEM Research Articles

An optimal adaptive finite element method for elastoplasticity

An adaptive finite element algorithm for problems in elastoplasticity with hardening is proven to be of optimal convergence with respect to the notion of approximation classes. The results rely on the equivalence of the errors of the stresses and energies resulting from Jensen's inequality. Numerical experiments study the influence of the hardening and bulk parameters to the convergence behavior of the AFEM algorithm. This is the first optimal adaptive FEM for a variational inequality.

Adaptive concurrent multi-scale FEM for trans-scale damage evolution in heterogeneous concrete

This paper presents a new adaptive concurrent multi-scale FEM (ACMFEM) to simulate the trans-scale failure processes from material damage in meso-scale to local failure area in macro-scale and eventually to global failure for concrete structures. The method can carry out automatically trans-scale coupling analysis on the failure process without user intervention. As cases studies of the method, the trans-scale failure processes of a L-shaped concrete component subjected to gradually increasing displacement loading and reinforced concrete (RC) column under seismic loading are simulated by using the developed method. The results show that, the developed method can be used to reveal the trans-scale failure mechanism of concrete structures by considering the process from material damage in meso-scale to local failure area in macro-scale and eventually to global failure in structural scale; and it is reliable in simulation on evolving damage and failure in concrete structures with the adaptive capability as well as better computational efficiency.

Ordering of Elements for the Volume & Neighbors Algorithm Constructing Elimination Trees for 2D and 3D h- adaptive FEM

In this paper we analyze the optimality of the volume and neighbors algorithm constructing elimination trees for three dimensional h-adaptive finite element method codes. The algorithm is a greedy algorithm that constructs the elimination trees based on the bottom up analysis of the computational mesh. We compare the results of the volume and neighbors greedy algorithm with the global dynamic programming optimization performed on a class of elimination trees. The comparison is based on the Directed Acyclic Graph (DAG) constructed for model grids. We construct DAGs for two model grids: a two dimensional grid refined towards point singularitiy and a two dimensional grid refined towards edge singularity. We show that the quasi-optimal trees created by the volume and neighbors algorithm for the model grids are also captured by the dynamic programming procedure. It means that created elimination trees are optimal in the considered class of elimination trees. We show that different element orderings at the input of the volume and neighbors algorithm result in different computational costs of the multi-frontal solver algorithm executed over the resulting elimination trees. Finally we present the ordering of elements that results in optimal (in the considered class) elimination trees. The theoretical results are verified with numerical experiments performed on a three dimensional grids with point, edge and face singularities.

Constant-free explicit error estimator with sharp upper error bound property for adaptive FE analysis in elasticity and fracture

SUMMARY The explicit residual-based error estimator originally proposed by Babuska and Miller (1987) for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators. Indeed, it provides a theoretically guaranteed upper bound on a discretization error, measured in the energy norm, and requires small (nearly negligible) post-processing computational effort. The main issue with this classical estimator, however, is that an upper error bound is explicitly computable up to an unknown multiplicative constant. In this paper, we track the source of this generic constant, revise the original derivation procedure and derive analytically four pre-computable constants that constitute an upper error bound, resulting in, what we then call, the constant-free error estimator of the Babuska-Miller type. The performance of this constant-free estimator, as well as an adaptive FEM based on it, are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation. The special property of the estimator is a sharp upper bound of the error: effectivity indices in the range of 1.2–2.0 are obtained, what is treated as practically (very) acceptable. In terms of efficiency judged against simplicity and inexpensiveness, the proposed constant-free explicit estimator is superior to, for example, corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known, yet more intricate and complex error estimation techniques. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical simulations of arteries with an adaptive finite element method

This paper presents applications of the adaptive FEM in computational biomechanics. The study involves finite deformation of artery walls built of layers of elastic, nearly incompressible (rubber-like) materials. The models are reinforced with nearly inextensible collagen fibres stretched along the directions close to the circumferential direction. A Total Lagrangian technique based on the displacement-pressure formulation is presented. Residual error estimates and finite element mesh adaptation strategies are developed. The procedures are illustrated for solutions of arteries under various load considerations.

L2and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces

Surface Finite Element Methods (SFEM) are widely used to solve surface partial differential equations arising in applications including crystal growth, fluid mechanics and computer graphics. A posteriori error estimators are computable measures of the error and are used to implement adaptive mesh refinement. Previous studies of a posteriori error estimation in SFEM have mainly focused on bounding energy norm errors. In this work we derive a posteriori L2 and pointwise error estimates for piecewise linear SFEM for the Laplace-Beltrami equation on implicitly defined surfaces. There are two main error sources in SFEM, a “Galerkin error” arising in the usual way for finite element methods, and a “geometric error” arising from replacing the continuous surface by a discrete approximation when writing the finite element equations. Our work includes numerical estimation of the dependence of the error bounds on the geometric properties of the surface. We provide also numerical experiments where the estimators have been used to implement an adaptive FEM over surfaces with different curvatures.

An execution time and energy model for an energy-aware execution of a conjugate gradient method with CPU/GPU collaboration

The parallel preconditioned conjugate gradient method (CGM) is used in many applications of scientific computing and often has a critical impact on their performance and energy consumption. This article investigates the energy-aware execution of the CGM on multi-core CPUs and GPUs used in an adaptive FEM. Based on experiments, an application-specific execution time and energy model is developed. The model considers the execution speed of the CPU and the GPU, their electrical power, voltage and frequency scaling, the energy consumption of the memory as well as the time and energy needed for transferring the data between main memory and GPU memory. The model makes it possible to predict how to distribute the data to the processing units for achieving the most energy efficient execution: the execution might deploy the CPU only, the GPU only or both simultaneously using a dynamic and adaptive collaboration scheme. The dynamic collaboration enables an execution minimising the execution time. By measuring execution times for every FEM iteration, the data distribution is adapted automatically to changing properties, e.g. the data sizes.

Numerical Analysis of an Adaptive FEM for Distributed Flux Reconstruction

AbstractThis paper studies convergence analysis of an adaptive finite element algorithm for numerical estimation of some unknown distributed flux in a stationary heat conduction system, namely recovering the unknown Neumann data on interior inaccessible boundary using Dirichlet measurement data on outer accessible boundary. Besides global upper and lower bounds established in [23], a posteriori local upper bounds and quasi-orthogonality results concerning the discretization errors of the state and adjoint variables are derived. Convergence and quasi-optimality of the proposed adaptive algorithm are rigorously proved. Numerical results are presented to illustrate the quasi-optimality of the proposed adaptive method.

Multilevel Adaptive Algorithm for Multiscale Analysis of Heterogeneous Materials

AbstractA multilevel adaptive FEM is proposed for multiscale mechanical analysis of heterogeneous materials. The algorithm is developed based on the extended multiscale FEM (EMSFEM), which has been proven to be an efficient method for multiscale mechanical analysis. A residual force-based error estimate is introduced, which is able to detect both global and local errors economically. The concept of microscopic numerical base function (NBF) is introduced, which can be used to refine the local results at the regions with a high-stress gradient level by level. Two efficient methods for updating the microscopic NBFs were developed, i.e., the hierarchical NBFs scheme and the residual force NBFs scheme. A hierarchical linear solver is designed to solve the multilevel equation system in each iteration step. In the algorithm, the calculations at the current iteration step take full advantage of the results of previous iteration steps, which is helpful to save computational resources. The numerical examples demons...

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear non-symmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\r{a}rding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

Adaptive stochastic Galerkin FEM

A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain.An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established.Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients.Details on the implementation with the open-source software framework ALEA are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem evaluated in the FEniCS environment.Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.

Adaptive Frequency-Dependent Shape Functions for Accurate Estimation of Modal Frequencies

AbstractAn adaptive linear FEM for modal analysis of structures is presented in this paper. The method uses frequency-dependent shape functions in addition to conventional low-order polynomial interpolation functions to improve accuracy in natural-frequency and mode-shape estimation. The proposed technique requires the iterative computation of the mass and stiffness matrices because shape functions are adjusted recursively as functions of estimated natural frequencies without mesh adaptation or refinement. Applied to frame structures, the method uses conventional polynomial interpolating functions for axial, bending, and torsional displacements and additional generalized coordinates multiplied by frequency-dependent functions derived from modal analysis of continuous beam models. Because these additional functions or their derivatives do not vanish at nodes located at the element boundaries, linear kinematic constraints are imposed to the augmented set of displacement coordinates to ensure displacement fi...

Hp-FEM electromechanical transduction model of ionic polymer–metal composites

We propose a mathematically explicit model of ionic polymer–metal composite (IPMC) materials that consist of a coupled system of Poisson–Nernst–Planck–Euler equations, discretized by means of adaptive higher-order finite element methods (hp-FEM). We show that due to the transient character of the problem and different fields in terms of the locations of field gradients in the domain it is efficient to use adaptive algorithms that are capable of changing the mesh dynamically in time. We also show that due to large qualitative and quantitative differences between the four solution components–cation concentration, electric potential, x-directional displacement, and y-directional displacement–it is efficient to approximate them on different meshes using a novel adaptive multi-mesh hp-FEM. The study is accompanied with computations and comparisons of the adaptive multi-mesh hp-FEM with a number of adaptive FEM algorithms.

The [formula omitted]-adaptive FEM based on continuous Sobolev embeddings: Isotropic refinements

The aim of this paper is to present a new class of smoothness testing strategies in the context of hp-adaptive refinements based on continuous Sobolev embeddings. In addition to deriving a modified form of the 1d smoothness indicators introduced in [26], they will be extended and applied to a higher dimensional framework. A few numerical experiments in the context of the hp-adaptive FEM for a linear elliptic PDE will be performed.

A numerical method to estimate temperature intervals for transient convection–diffusion heat transfer problems

This paper presents a numerical method to estimate intervals of temperature for transient convection–diffusion heat transfer problems when uncertainty of thermal parameters is characterized by the interval. A deterministic relationship of interval variables between temperature and thermal parameters is setup by Taylor series expansion and the interval analysis, and the lower and upper bounds of temperature can be estimated by a temporally piecewise adaptive algorithm and FEM. A prescribed computing accuracy at each discretized time interval can be achieved via an adaptive process for different size of time step, thus the computing accuracy over the whole time domain can be maintained. A 2D numerical example is provided to verify the proposed approach, and a good accordance can be observed in the comparison of results given by the combinatorial, probability and proposed approaches. The impact of the order of Taylor expansion and the size of time step on the result is discussed.

EachH1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd

We consider the solution of second order elliptic PDEs in $\R^d$ with inhomogeneous Dirichlet data by means of an $h$-adaptive FEM with fixed polynomial order $p\in\N$. As model example serves the Poisson equation with mixed Dirichlet-Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an $H^{1/2}$-stable projection, for instance, the $L^2$-projection for $p=1$ or the Scott-Zhang projection for general $p\ge1$. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each $H^{1/2}$-stable projection yields convergence of the adaptive algorithm even with quasi-optimal convergence rate. Numerical experiments with the $L^2$- and Scott-Zhang projection conclude the work.

Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data

We consider the solution of a second order elliptic PDE with inhomogeneous Dirichlet data by means of adaptive lowest-order FEM. As is usually done in practice, the given Dirichlet data are discretized by nodal interpolation. As model example serves the Poisson equation with mixed Dirichlet–Neumann boundary conditions. For error estimation, we use an edge-based residual error estimator which replaces the volume residual contributions by edge oscillations. For 2D, we prove convergence of the adaptive algorithm even with optimal convergence rate. For 2D and 3D, we show convergence if the nodal interpolation operator is replaced by the L2-projection or the Scott–Zhang quasi-interpolation operator. As a byproduct of the proof, we show that the Scott–Zhang operator converges pointwise to a limiting operator as the mesh is locally refined. This property might be of independent interest besides the current application. Finally, numerical experiments conclude the work.

Convergence of adaptive FEM for some elliptic obstacle problem

In this work, we treat the convergence of adaptive lowest-order FEM for some elliptic obstacle problem with affine obstacle. For error estimation, we use a residual error estimator from [D. Braess, C. Carstensen, and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), pp. 455–471]. We extend recent ideas from [J. Cascon, C. Kreuzer, R. Nochetto, and K. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), pp. 2524–2550] for the unrestricted variational problem to overcome the lack of Galerkin orthogonality. The main result states that an appropriately weighted sum of energy error, edge residuals and data oscillations satisfies a contraction property within each step of the adaptive feedback loop. This result is superior to a prior result from Braess et al. (2007) in two ways: first, it is unnecessary to control the decay of the data oscillations explicitly; second, our analysis avoids the use of some discrete local efficiency estimate so that the local mesh-refinement is fairly arbitrary.

Adaptive reduced basis finite element heterogeneous multiscale method

An adaptive reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is proposed for elliptic problems with multiple scales. The multiscale method is based on the RB-FE-HMM introduced in [A. Abdulle, Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys. 231 (21) (2012) 7014–7036]. It couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains. Unlike classical numerical homogenization methods, the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions (the reduced basis) obtained by a greedy algorithm in an offline stage. In this paper we present a residual-based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest. For both type of adaptive strategies, rigorous a posteriori error estimates are derived and corresponding error estimators are proposed. In contrast to the adaptive finite element heterogeneous multiscale method (FE-HMM), there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement. Up to an offline preliminary stage, the RB-FE-HMM has the same computational complexity as a standard adaptive FEM for the effective problem. Two and three dimensional numerical experiments confirm the efficiency of the RB-FE-HMM and illustrate the improvements compared to the adaptive FE-HMM.

Dynamic Distribution of Workload between CPU and GPU for a Parallel Conjugate Gradient Method in an Adaptive FEM

The parallel preconditioned conjugate gradient method (CGM) is often used in adaptive FEMs and has a critical impact on the performance. This article proposes a method for dynamically balancing the computational load of this CGM between CPU and GPU. For the determination of the optimal balance of the computational load on CPU and GPU, an execution time model for the CGM is developed which considers the different execution speeds of the two kinds of processing units. The model relies on data-specific and machine-specific parameters which are both determined at runtime. The accuracy of the model is verified in experiments. This auto-tuning-based approach for CPU/GPU collaboration enables significant performance benefits compared to CPU-only or GPU-only execution.