Strict bounding of quantities of interest in computations based on domain decomposition

This paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods. The proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems : on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver. Beside practical considerations on the parallel computation of the bounds, it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation. Academic assessments are given on 2D static linear mechanic problems.Comment: Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, online previe

Assessment of the local stress state through macroscopic variables

Macroscopic quantities beyond effective elastic tensors are presented that can be used to assess the local state of stress within a composite in the linear elastic regime. These are presented in a general homogenization context. It is shown that the gradient of the effective elastic property can be used to develop a lower bound on the maximum pointwise equivalent stress in the fine-scale limit. Upper bounds are more sensitive and are correlated with the distribution of states of the equivalent stress in the fine-scale limit. The upper bounds are given in terms of the macrostress modulation function. This function gauges the magnitude of the actual stress. For l ≤ p \u3c ∞, upper bounds are found on the limit superior of the sequence of Lp norms of stresses associated with discrete microstructure in the fine-scale limit. Conditions are given for which upper bounds can be found on the limit superior of the sequence of L∞ norms of stresses associated with the discrete microstructure in the fine-scale limit. For microstructure with oscillation on a sufficiently small scale we are able to give pointwise bounds on the actual stress in terms of the macrostress modulation function

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

More and more challenging designs are required everyday in today¿s industries. The traditional trial and error procedure commonly used for mechanical parts design is not valid any more since it slows down the design process and yields suboptimal designs. For structural components, one alternative consists in using shape optimization processes which provide optimal solutions. However, these techniques require a high computational effort and require extremely efficient and robust Finite Element (FE) programs. FE software companies are aware that their current commercial products must improve in this sense and devote considerable resources to improve their codes. In this work we propose to use the Cartesian Grid Finite Element Method, cgFEM as a tool for efficient and robust numerical analysis. The cgFEM methodology developed in this thesis uses the synergy of a variety of techniques to achieve this purpose, but the two main ingredients are the use of Cartesian FE grids independent of the geometry of the component to be analyzed and an efficient hierarchical data structure. These two features provide to the cgFEM technology the necessary requirements to increase the efficiency of the cgFEM code with respect to commercial FE codes. As indicated in [1, 2], in order to guarantee the convergence of a structural shape optimization process we need to control the error of each geometry analyzed. In this sense the cgFEM code also incorporates the appropriate error estimators. These error estimators are specifically adapted to the cgFEM framework to further increase its efficiency. This work introduces a solution recovery technique, denoted as SPR-CD, that in combination with the Zienkiewicz and Zhu error estimator [3] provides very accurate error measures of the FE solution. Additionally, we have also developed error estimators and numerical bounds in Quantities of Interest based on the SPR-CD technique to allow for an efficient control of the quality of the numerical solution. Regarding error estimation, we also present three new upper error bounding techniques for the error in energy norm of the FE solution, based on recovery processes. Furthermore, this work also presents an error estimation procedure to control the quality of the recovered solution in stresses provided by the SPR-CD technique. Since the recovered stress field is commonly more accurate and has a higher convergence rate than the FE solution, we propose to substitute the raw FE solution by the recovered solution to decrease the computational cost of the numerical analysis. All these improvements are reflected by the numerical examples of structural shape optimization problems presented in this thesis. These numerical analysis clearly show the improved behavior of the cgFEM technology over the classical FE implementations commonly used in industry.Cada d'¿a dise¿nos m'as complejos son requeridos por las industrias actuales. Para el dise¿no de nuevos componentes, los procesos tradicionales de prueba y error usados com'unmente ya no son v'alidos ya que ralentizan el proceso y dan lugar a dise¿nos sub-'optimos. Para componentes estructurales, una alternativa consiste en usar procesos de optimizaci'on de forma estructural los cuales dan como resultado dise¿nos 'optimos. Sin embargo, estas t'ecnicas requieren un alto coste computacional y tambi'en programas de Elementos Finitos (EF) extremadamente eficientes y robustos. Las compa¿n'¿as de programas de EF son conocedoras de que sus programas comerciales necesitan ser mejorados en este sentido y destinan importantes cantidades de recursos para mejorar sus c'odigos. En este trabajo proponemos usar el M'etodo de Elementos Finitos basado en mallados Cartesianos (cgFEM) como una herramienta eficiente y robusta para el an'alisis num'erico. La metodolog'¿a cgFEM desarrollada en esta tesis usa la sinergia entre varias t'ecnicas para lograr este prop'osito, cuyos dos ingredientes principales son el uso de los mallados Cartesianos de EF independientes de la geometr'¿a del componente que va a ser analizado y una eficiente estructura jer'arquica de datos. Estas dos caracter'¿sticas confieren a la tecnolog'¿a cgFEM de los requisitos necesarios para aumentar la eficiencia del c'odigo cgFEM con respecto a c'odigos comerciales. Como se indica en [1, 2], para garantizar la convergencia del proceso de optimizaci'on de forma estructural se necesita controlar el error en cada geometr'¿a analizada. En este sentido el c'odigo cgFEM tambi'en incorpora los apropiados estimadores de error. Estos estimadores de error han sido espec'¿ficamente adaptados al entorno cgFEM para aumentar su eficiencia. En esta tesis se introduce un proceso de recuperaci'on de la soluci'on, llamado SPR-CD, que en combinaci'on con el estimador de error de Zienkiewicz y Zhu [3], da como resultado medidas muy precisas del error de la soluci'on de EF. Adicionalmente, tambi'en se han desarrollado estimadores de error y cotas num'ericas en Magnitudes de Inter'es basadas en la t'ecnica SPR-CD para permitir un eficiente control de la calidad de la soluci'on num'erica. Respecto a la estimaci'on de error, tambi'en se presenta un proceso de estimaci'on de error para controlar la calidad del campo de tensiones recuperado obtenido mediante la t'ecnica SPR-CD. Ya que el campo recuperado es por lo general m'as preciso y tiene un mayor orden de convergencia que la soluci'on de EF, se propone sustituir la soluci'on de EF por la soluci'on recuperada para disminuir as'¿ el coste computacional del an'alisis num'erico. Todas estas mejoras se han reflejado en esta tesis mediante ejemplos num'ericos de problemas de optimizaci'on de forma estructural. Los resultados num'ericos muestran claramente un mejor comportamiento de la tecnolog'¿a cgFEM con respecto a implementaciones cl'asicas de EF com'unmente usadas en la industria.Nadal Soriano, E. (2014). Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/35620TESI

Accurate recovery-based upper error bounds for the extended finite element framework

This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests

Non-regularised inverse finite element analysis for 3D traction force microscopy

The tractions that cells exert on a gel substrate from the observed displacements is an increasingly attractive and valuable information in biomedical experiments. The computation of these tractions requires in general the solution of an inverse problem. Here, we resort to the discretisation with finite elements of the associated direct variational formulation, and solve the inverse analysis using a least square approach. This strategy requires the minimisation of an error functional, which is usually regularised in order to obtain a stable system of equations with a unique solution. In this paper we show that for many common threedimensional geometries, meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational cost of the inverse problem becomes equivalent to a direct finite element problem. For the non-regularised functional, we deduce the necessary and sufficient conditions that the dimensions of the interpolated displacement and traction fields must preserve in order to exactly satisfy or yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to some illustrative examples and to real experimental data. Due to the relevance of the results for biologists and modellers, the article concludes with some practical rules that the finite element discretisation must satisfy.Peer ReviewedPostprint (author's final draft

On barrier and modified barrier multigrid methods for 3d topology optimization

One of the challenges encountered in optimization of mechanical structures, in particular in what is known as topology optimization, is the size of the problems, which can easily involve millions of variables. A basic example is the minimum compliance formulation of the variable thickness sheet (VTS) problem, which is equivalent to a convex problem. We propose to solve the VTS problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\ Polyak and later studied by Ben-Tal and Zibulevsky and others. The most computationally expensive part of the algorithm is the solution of linear systems arising from the Newton method used to minimize a generalized augmented Lagrangian. We use a special structure of the Hessian of this Lagrangian to reduce the size of the linear system and to convert it to a form suitable for a standard multigrid method. This converted system is solved approximately by a multigrid preconditioned MINRES method. The proposed PBM algorithm is compared with the optimality criteria (OC) method and an interior point (IP) method, both using a similar iterative solver setup. We apply all three methods to different loading scenarios. In our experiments, the PBM method clearly outperforms the other methods in terms of computation time required to achieve a certain degree of accuracy

Debonding of cellular structures with fibre-reinforced cell walls under shear deformation

Many natural structures are cellular solids at millimetre scale and fibre-reinforced composites at micrometer scale. For these structures, mechanical properties are associated with cell strength, and phenomena such as cells separation through debonding of the middle lamella in cell walls is key in explaining some important characteristics or behaviour. To explore such phenomena, we model cellular structures with nonlinear hyperelastic cell walls under large shear deformations, and incorporate cell wall material anisotropy and unilateral contact between neighbouring cells in our models. Analytically, we show that, for two cuboid walls in unilateral contact and subject to generalised shear, gaps can appear at the interface between the deforming walls. Numerically, when finite element models of periodic structures with hexagonal cells are sheared, significant cell separation is captured diagonally across the structure. Our analysis further reveals that separation is less likely between cells with high internal cell pressure (e.g. in fresh and growing fruit and vegetables) than between cells where the internal pressure is low (e.g. in cooked or ageing plants)

A sequential linear programming (SLP) approach for uncertainty analysis-based data-driven computational mechanics

In this article, an efficient sequential linear programming algorithm (SLP) for uncertainty analysis-based data-driven computational mechanics (UA-DDCM) is presented. By assuming that the uncertain constitutive relationship embedded behind the prescribed data set can be characterized through a convex combination of the local data points, the upper and lower bounds of structural responses pertaining to the given data set, which are more valuable for making decisions in engineering design, can be found by solving a sequential of linear programming problems very efficiently. Numerical examples demonstrate the effectiveness of the proposed approach on sparse data set and its robustness with respect to the existence of noise and outliers in the data set

Differential Evolution: A Tool for Global Optimization

This report describes a tool for global optimization that implements the Differential Evolution optimization algorithm as a new Excel add-in. The tool takes a step beyond Excel’s Solver add-in, because Solver often returns a local minimum, that is, a minimum that is less than or equal to nearby points, while Differential Evolution solves for the global minimum, which includes all feasible points. Despite complex underlying mathematics, the tool is relatively easy to use, and can be applied to practical optimization problems, such as establishing pricing and awards in a hotel loyalty program. The report demonstrates an example of how to develop an optimum approach to that problem