A posteriori modeling error estimates in the optimization of two-scale elastic composite materials

The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern

Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. We explore the capabilities of a recovery technique based on an enhanced MLS fitting, which directly provides continuous interpolated fields, to obtain estimates of the error in energy norm as an alternative to the superconvergent patch recovery (SPR). Boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results show the high accuracy of the proposed error estimator

Model adaptivity for finite element analysis of thin or thick plates based on equilibrated boundary stress resultants

Purpose – The purpose of this paper is to address error-controlled adaptive finite element (FE) method for thin and thick plates. A procedure is presented for determining the most suitable plate model (among available hierarchical plate models) for each particular FE of the selected mesh, that is provided as the final output of the mesh adaptivity procedure. \ud \ud Design/methodology/approach – The model adaptivity procedure can be seen as an appropriate extension to model adaptivity for linear elastic plates of so-called equilibrated boundary traction approach error estimates, previously proposed for 2D/3D linear elasticity. Model error indicator is based on a posteriori element-wise computation of improved (continuous) equilibrated boundary stress resultants, and on a set of hierarchical plate models. The paper illustrates the details of proposed model adaptivity procedure for choosing between two most frequently used plate models: the one of Kirchhoff and the other of Reissner-Mindlin. The implementation details are provided for a particular case of the discrete Kirchhoff quadrilateral four-node plate FE and the corresponding Reissner-Mindlin quadrilateral with the same number of nodes. The key feature for those elements that they both provide the same quality of the discretization space (and thus the same discretization error) is the one which justifies uncoupling of the proposed model adaptivity from the mesh adaptivity. \ud \ud Findings – Several numerical examples are presented in order to illustrate a very satisfying performance of the proposed methodology in guiding the final choice of the optimal model and mesh in analysis of complex plate structures. \ud \ud Originality/value – The paper confirms that one can make an automatic selection of the most appropriate plate model for thin and thick plates on the basis of proposed model adaptivity procedure.\u

Error estimation and adaptivity for incompressible, non–linear (hyper–)elasticity

A Galerkin finite element method is developed for non–linear, incompressible (hyper) elasticity, and a posteriori error estimates are derived for both linear functionals of the solution and linear functionals of the stress on a boundary where Dirichlet boundary conditions are applied. A second, higher order method for calculating a linear functional of the stress on a Dirichlet boundary is also presented together with an a posteriori error estimator for this approach. An implementation for a 2D model problem with known solution demonstrates the accuracy of the error estimators. Finally the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity

A Variational r-Adaption and Shape-Optimization Method for Finite-Deformation Elasticity

This paper is concerned with the formulation of a variational r-adaption method for finite-deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configuration forces for isoparametric elements and nonlinear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi-infinite crack in linear and nonlinear elastic bodies; and the optimization of the shape of elastic inclusions

Real-time Error Control for Surgical Simulation

Objective: To present the first real-time a posteriori error-driven adaptive finite element approach for real-time simulation and to demonstrate the method on a needle insertion problem. Methods: We use corotational elasticity and a frictional needle/tissue interaction model. The problem is solved using finite elements within SOFA. The refinement strategy relies upon a hexahedron-based finite element method, combined with a posteriori error estimation driven local $h$-refinement, for simulating soft tissue deformation. Results: We control the local and global error level in the mechanical fields (e.g. displacement or stresses) during the simulation. We show the convergence of the algorithm on academic examples, and demonstrate its practical usability on a percutaneous procedure involving needle insertion in a liver. For the latter case, we compare the force displacement curves obtained from the proposed adaptive algorithm with that obtained from a uniform refinement approach. Conclusions: Error control guarantees that a tolerable error level is not exceeded during the simulations. Local mesh refinement accelerates simulations. Significance: Our work provides a first step to discriminate between discretization error and modeling error by providing a robust quantification of discretization error during simulations.Comment: 12 pages, 16 figures, change of the title, submitted to IEEE TBM