A multi-space error estimation approach for meshfree methods

summary:Error-controlled adaptive meshfree methods are presented for both global error measures, such as the energy norm, and goal-oriented error measures in terms of quantities of interest. The meshfree method chosen in this paper is the reproducing kernel particle method (RKPM), since it is based on a Galerkin scheme and therefore allows extensions of quality control approaches as already developed for the finite element method. Our approach of goal-oriented error estimation is based on the well-established technique using an auxiliary dual problem. To keep the formulation general and to add versatility, a multi-space approach is used, where the dual problem is solved numerically using a different approximation space than the one employed in the associated primal problem. This can be realized with meshfree methods at no additional cost. Possible merits of this multi-space approach are discussed and an illustrative numerical example is presented

Pre- and postprocessing techniques for determining goodness of computational meshes

Research in error estimation, mesh conditioning, and solution enhancement for finite element, finite difference, and finite volume methods has been incorporated into AUDITOR, a modern, user-friendly code, which operates on 2D and 3D unstructured neutral files to improve the accuracy and reliability of computational results. Residual error estimation capabilities provide local and global estimates of solution error in the energy norm. Higher order results for derived quantities may be extracted from initial solutions. Within the X-MOTIF graphical user interface, extensive visualization capabilities support critical evaluation of results in linear elasticity, steady state heat transfer, and both compressible and incompressible fluid dynamics

Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method

We discuss, in this paper, a common ux-free method for the computation of strict error bounds for linear and nonlinear Finite Element computations. In the linear case, the error bounds are on the energy norm of the error, while, in the nonlinear case, the concept of error in constitutive relation is used. In both cases, the error bounds are strict in the sense that they re- fer to the exact solution of the continuous equations, rather than to some FE computation over a refined mesh. For both linear and nonlinear solid mechanics, this method is based on the computation of a statically admissible stress field, which is performed as a series of local problems on patches of elements. There is no requirement to solve a previous problem of ux equilibration globally, as happens with other methods.Postprint (published version

Realistic computable error bounds for three dimensional finite element analyses in linear elasticity

We obtain a computable estimator for the energy norm of the error in piecewise affine and piecewise quadratic finite element approximations of linear elasticity in three dimensions. We show that the estimator provides guaranteed upper bounds on the energy norm of the error as well as (up to a constant and data oscillation terms) local lower bounds

A hp-adaptive discontinuous Galerkin finite element method for accurate configurational force brittle crack propagation

Engineers require accurate determination of the configurational force at the crack tip for fracture fatigue analysis and accurate crack propagation. However, obtain- ing highly accurate crack tip configuration force values is challenging with numer- ical methods requiring knowledge of the stress field around the crack tip a priori. In this thesis, the symmetric interior penalty discontinuous Galerkin finite element method is combined with a residual based a posteriori error estimator which drives a hp-adaptive mesh refinement scheme to determine accurate solutions of the stress field about about the crack. This facilitates the development of a novel method to calculate the crack tip configurational force that is accurate, requires no a priori knowledge of the stress field about the crack tip with, its error bound by an error estimator which is calculated a posteriori. Benchmark values of the crack tip con- figurational force are presented for problems containing multiple mixed mode cracks in both isotropic and anisotropic materials. Additionally, the hp-adaptivity is com- bined with a mathematical analysis of the stress field at the crack tip to critique the convergence and limitations of other methods in the literature to calculate the crack tip configurational force. Two methods for staggered quasi-static crack prop- agation are also presented. An rp-adaptive method which is simple to implement and computationally inexpensive, element edges aligned with the crack propagation path with the exploitation of the discontinuous Galerkin edge sti↵ness terms exist- ing along element interfaces to propagate a crack. The second method is denoted the hpr-adaptive method which combines the accurate computation of the crack tip configuration force with r-adaptivity to produce a computationally expensive but accurate method to propagate multiple cracks simultaneously. Further, for indeter- minate systems, an average boundary condition that restrains rigid body motion and rotation is introduced to make the system determinate

Improved recovery of admissible stress in domain decomposition methods - application to heterogeneous structures and new error bounds for FETI-DP

This paper investigates the question of the building of admissible stress field in a substructured context. More precisely we analyze the special role played by multiple points. This study leads to (1) an improved recovery of the stress field, (2) an opportunity to minimize the estimator in the case of heterogeneous structures (in the parallel and sequential case), (3) a procedure to build admissible fields for FETI-DP and BDDC methods leading to an error bound which separates the contributions of the solver and of the discretization