Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method

A variety of meshless methods have been developed in the last fifteen years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The main objective of this thesis is the development of an efficient and accurate algorithm based on meshless methods for the solution of problems involving both material and geometrical nonlinearities, which are of practical importance in many engineering applications, including geomechanics, metal forming and biomechanics. One of the most commonly used meshless methods, the element-free Galerkin method (EFGM) is used in this research, in which maximum entropy shape functions (max-ent) are used instead of the standard moving least squares shape functions, which provides direct imposition of the essential boundary conditions. Initially, theoretical background and corresponding computer implementations of the EFGM are described for linear and nonlinear problems. The Prandtl-Reuss constitutive model is used to model elasto-plasticity, both updated and total Lagrangian formulations are used to model finite deformation and consistent or algorithmic tangent is used to allow the quadratic rate of asymptotic convergence of the global Newton-Raphson algorithm. An adaptive strategy is developed for the EFGM for two- and three-dimensional nonlinear problems based on the Chung & Belytschko error estimation procedure, which was originally proposed for linear elastic problems. A new FE-EFGM coupling procedure based on max-ent shape functions is proposed for linear and geometrically nonlinear problems, in which there is no need of interface elements between the FE and EFG regions or any other special treatment, as required in the most previous research. The proposed coupling procedure is extended to become adaptive FE-EFGM coupling for two- and three-dimensional linear and nonlinear problems, in which the Zienkiewicz & Zhu error estimation procedure with the superconvergent patch recovery method for strains and stresses recovery are used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region of the problem domain. Parallel computer algorithms based on distributed memory parallel computer architecture are also developed for different numerical techniques proposed in this thesis. In the parallel program, the message passing interface library is used for inter-processor communication and open-source software packages, METIS and MUMPS are used for the automatic domain decomposition and solution of the final system of linear equations respectively. Separate numerical examples are presented for each algorithm to demonstrate its correct implementation and performance, and results are compared with the corresponding analytical or reference results

Recovery type a posteriori error estimation of an adaptive finite element method for Cahn--Hilliard equation

In this paper, we derive a novel recovery type a posteriori error estimation of the Crank-Nicolson finite element method for the Cahn--Hilliard equation. To achieve this, we employ both the elliptic reconstruction technique and a time reconstruction technique based on three time-level approximations, resulting in an optimal a posteriori error estimator. We propose a time-space adaptive algorithm that utilizes the derived a posteriori error estimator as error indicators. Numerical experiments are presented to validate the theoretical findings, including comparing with an adaptive finite element method based on a residual type a posteriori error estimator.Comment: 36 pages, 7 figure

Recovery Techniques For Finite Element Methods And Their Applications

Recovery techniques are important post-processing methods to obtain improved approximate solutions from primary data with reasonable cost. The practical us- age of recovery techniques is not only to improve the quality of approximation, but also to provide an asymptotically exact posteriori error estimators for adaptive meth- ods. This dissertation presents recovery techniques for nonconforming finite element methods and high order derivative as well as applications of gradient recovery. Our first target is to develop a systematic gradient recovery technique for Crouzeix- Raviart element. The proposed method uses finite element solution to build a better approximation of the exact gradient based on local least square fittings. Due to poly- nomial preserving property of least square fitting, it is easy to show that the new proposed method preserves quadratic polynomials. In addition, the proposed gra- dient recovery is linearly bounded. Numerical tests indicate the recovered gradient is superconvergent to the exact gradient for both second order elliptic equation and Stokes equation. The gradient recovery technique can be used in a posteriori error estimates for Crouzeix-Raviart element, which is relatively simple to implement and problem independent. Our second target is to propose and analyze a new effective Hessian recovery for continuous finite element of arbitrary order. The proposed Hessian recovery is based on polynomial preserving recovery. The proposed method preserves polynomials of degree (k + 1) on general unstructured meshes and polynomials of degree (k + 2) on translation invariant meshes. Based on it polynomial preserving property, we can able to prove superconvergence of the proposed method on mildly structured meshes. In addition, we establish the ultraconvergence result for the new Hessian recovery technique on translation invariant finite element space of arbitrary order. Our third target is to demonstrate application of gradient recovery in eigenvalue computation. We propose two superconvergent two-grid methods for elliptic eigen- value problems by taking advantage of two-gird method, two-space method, shifted- inverse power method, and gradient recovery enhancement. Theoretical and numer- ical results reveal that the proposed methods provide superconvergent eigenfunction approximation and ultraconvergent eigenvalue approximation. In addition, two mul- tilevel adaptive methods based recovery type a posterior error estimate are proposed

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed

h-Adaptive finite element method: extension of the isotropic error density recovery remeshing strategy of quadratic order

Orientador: Prof. Dr. Jucélio Tomás PereiraDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Mecânica. Defesa : Curitiba, 05/07/2018Inclui referências: p.86-90Área de concentração: Mecânica dos Sólidos e VibraçõesResumo: O Método de Elementos Finitos (MEF) é uma técnica para resolver numericamente problemas físicos comumente utilizada na engenheria. Um fator importante na obtenção de uma solução precisa e eficiente decorre da utilização adequada da malha de discretização. Tipicamente, técnicas h-adaptativas são empregadas para projeção de uma malha ótima, onde o erro estimado em cada elemento é distribuído e minimizado de acordo com um critério de malha ótima. Neste contexto, o presente trabalho estende e avalia o método de refino hadaptativo denominado de Recuperação da Densidade do Erro Isotrópica (IEDR) para elementos triangulares quadráticos. Inicialmente desenvolvida para elementos lineares, esta técnica baseia-se na recuperação de uma função densidade do erro em energia em conjunto com a solução de um problema de otimização que busca o tamanho do novo elemento. Dessa maneira, a metodologia IEDR aborda os erros provenientes do MEF de maneira que contenha informações locais com maior abrangência, já que, nesta metodologia, uma função densidade do erro é recuperada. Os parâmetros de qualidade de malha, obtidos através desta técnica, são comparados à tradicionais técnicas de projeto de malha denominada de Chp e à técnica Li- Bettess (LB). A estimativa dos erros de discretização é realizada através do estimador de erro a posteriori baseado em recuperação, onde os gradientes recuperados são obtidos pelo método Superconvergente de Recuperação de Padrões (Superconvergent Patch Recovery - SPR). A implementação computacional é elaborada no software Matlab®, sendo a geração de malha realizada pelo gerador Bidimensional Anisotropic Mesh Generator (BAMG). Resultados numéricos demonstram que o processo h-adaptativo baseado na técnica IEDR obtém malhas convergentes para problemas com e sem singularidade, as quais apresentam, em geral, vantagens em relação ao número de graus de liberdade, à convergência e aos parâmetros de malha em comparação à tradicional técnica Chp e vantagens comparada à técnica LB para elementos quadráticos. Palavras-chave: Elemento Triangular de Deformações Lineares. h-adaptividade. Método dos Elementos Finitos. Estimadores de erro a posteriori. Recuperação da Densidade do Erro Isotrópica.Abstract: The finite element method (FEM) is a technique used to numerically solve physics problems which is often used in engineering. One factor in obtaining a solution that has acceptable accuracy is using adequate mesh discretization. Typically, h-adaptive techniques are used to determine new element sizes based on errors distributed among each element following an optimum mesh criterion. In this context, the current work proposes to extend and analyze the Isotropic Error Density Recovery (IEDR) h-refinement method for quadratic triangular finite elements. Initially developed for linear triangular finite elements, the extended technique is based on the recovery of an error density function, such that an optimization technique is used to search for the new element sizes. Hence, the IEDR technique utilizes more information of the local errors to design element sizes due to the recovery of an element error density function. The h-adaptive finite element method process based on the IEDR technique is compared to the traditionally used Chp and Li-Bettess mesh design techniques found in the literature. The discretization error estimates are achieved via a recovery based a posteriori error estimator, whereas the recovered gradients are obtained using the Superconvergent Patch Recovery Method. The algorithm is implemented using Matlab®, while the mesh generation is done by the Bidimensional Anisotropic Mesh Generator (BAMG). Results show, overall, that the meshes designed through the proposed methodology obtain superior mesh quality parameters, less degrees of freedom and better convergence in comparison with the traditional Chp remeshing methodology and advantages compared to the Li-Bettess element size estimation technique for quadratic elements. Keywords: Linear Strain Triangle. h-adaptativity. Finite Element Method. a posteriori Error Estimates. Isotropic Error Density Recovery

Non-axisymmetric instabilities in self-gravitating tori around black holes, and solving Einstein constraints with superconvergent finite element methods

This thesis contains results on two related projects. In the first project, we explore non-axisymmetric instabilities in general relativistic accretion disks around black holes. Such disks are created as transient structures in several astrophysical scenarios, including mergers of compact objects and core collapse of massive stars. These disks are suggested for the role of cenral engines of gamma-ray bursts. We address the stability of these objects against the runaway and non-axisymmetric instabilities in the three-dimensional hydrodynamical fully general relativistic treatment. We explore three slender and moderately slender disk models with varying disk-to-black hole mass ratio. None of the models that we consider develop the runaway instability during the time span of the simulations, despite large radial axisymmetric oscillations, induced in the disks by the initial data construction procedure. All models develop unstable non-axisymmetric modes on a dynamical timescale. In simulations with dynamical general relativistic treatment, we observe two distinct types of instabilities: the Papaloizou-Pringle instability and the so-called Intermediate instability. The development of the nonaxisymmetric mode with azimuthal number m=1 is enhanced by the outspiraling motion of the black hole. The overall picture of the unstable modes in our disk models is similar to the Newtonian case. In the second project, we experiment with solving the Einstein constraint equations using finite elements on semistructured triangulations of multiblock grids. We illustrate our approach with a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $\psi$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, we evolve the constructed initial data using a high order finite-differencing multi-block approach and extract gravitational waves from the numerical solution