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

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

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

Large-scale tree-based unfitted finite elements for metal additive manufacturing

This thesis addresses large-scale numerical simulations of partial differential equations posed on evolving geometries. Our target application is the simulation of metal additive manufacturing (or 3D printing) with powder-bed fusion methods, such as Selective Laser Melting (SLM), Direct Metal Laser Sintering (DMLS) or Electron-Beam Melting (EBM). The simulation of metal additive manufacturing processes is a remarkable computational challenge, because processes are characterised by multiple scales in space and time and multiple complex physics that occur in intricate three-dimensional growing-in-time geometries. Only the synergy of advanced numerical algorithms and high-performance scientific computing tools can fully resolve, in the short run, the simulation needs in the area. The main goal of this Thesis is to design a a novel highly-scalable numerical framework with multi-resolution capability in arbitrarily complex evolving geometries. To this end, the framework is built by combining three computational tools: (1) parallel mesh generation and adaptation with forest-of-trees meshes, (2) robust unfitted finite element methods and (3) parallel finite element modelling of the geometry evolution in time. Our numerical research is driven by several limitations and open questions in the state-of-the-art of the three aforementioned areas, which are vital to achieve our main objective. All our developments are deployed with high-end distributed-memory implementations in the large-scale open-source software project FEMPAR. In considering our target application, (4) temporal and spatial model reduction strategies for thermal finite element models are investigated. They are coupled to our new large-scale computational framework to simplify optimisation of the manufacturing process. The contributions of this Thesis span the four ingredients above. Current understanding of (1) is substantially improved with rigorous proofs of the computational benefits of the 2:1 k-balance (ease of parallel implementation and high-scalability) and the minimum requirements a parallel tree-based mesh must fulfil to yield correct parallel finite element solvers atop them. Concerning (2), a robust, optimal and scalable formulation of the aggregated unfitted finite element method is proposed on parallel tree-based meshes for elliptic problems with unfitted external contour or unfitted interfaces. To the author’s best knowledge, this marks the first time techniques (1) and (2) are brought together. After enhancing (1)+(2) with a novel parallel approach for (3), the resulting framework is able to mitigate a major performance bottleneck in large-scale simulations of metal additive manufacturing processes by powder-bed fusion: scalable adaptive (re)meshing in arbitrarily complex geometries that grow in time. Along the development of this Thesis, our application problem (4) is investigated in two joint collaborations with the Monash Centre for Additive Manufacturing and Monash University in Melbourne, Australia. The first contribution is an experimentally-supported thorough numerical assessment of time-lumping methods, the second one is a novel experimentally-validated formulation of a new physics-based thermal contact model, accounting for thermal inertia and suitable for model localisation, the so-called virtual domain approximation. By efficiently exploiting high-performance computing resources, our new computational framework enables large-scale finite element analysis of metal additive manufacturing processes, with increased fidelity of predictions and dramatical reductions of computing times. It can also be combined with the proposed model reductions for fast thermal optimisation of the manufacturing process. These tools open the path to accelerate the understanding of the process-to-performance link and digital product design and certification in metal additive manufacturing, two milestones that are vital to exploit the technology for mass-production.Aquesta tesi tracta la simulació a gran escala d'equacions en derivades parcials sobre geometries variables. L'aplicació principal és la simulació de procesos de fabricació additiva (o impressió 3D) amb metalls i per mètodes de fusió de llit de pols, com ara Selective Laser Melting (SLM), Direct Metal Laser Sintering (DMLS) o Electron-Beam Melting (EBM). La simulació d'aquests processos és un repte computacional excepcional, perquè els processos estan caracteritzats per múltiples escales espaitemporals i múltiples físiques que tenen lloc sobre geometries tridimensionals complicades que creixen en el temps. La sinèrgia entre algorismes numèrics avançats i eines de computació científica d'alt rendiment és la única via per resoldre completament i a curt termini les necessitats en simulació d'aquesta àrea. El principal objectiu d'aquesta tesi és dissenyar un nou marc numèric escalable de simulació amb capacitat de multiresolució en geometries complexes i variables. El nou marc es construeix unint tres eines computacionals: (1) mallat paral·lel i adaptatiu amb malles de boscs d'arbre, (2) mètodes d'elements finits immersos robustos i (3) modelització en paral·lel amb elements finits de geometries que creixen en el temps. Algunes limitacions i problemes oberts en l'estat de l'art, que són claus per aconseguir el nostre objectiu, guien la nostra recerca. Tots els desenvolupaments s'implementen en arquitectures de memòria distribuïda amb el programari d'accés obert FEMPAR. Quant al problema d'aplicació, (4) s'investiguen models reduïts en espai i temps per models tèrmics del procés. Aquests models reduïts s'acoplen al nostre marc computacional per simplificar l'optimització del procés. Les contribucions d'aquesta tesi abasten els quatre punts de dalt. L'estat de l'art de (1) es millora substancialment amb proves riguroses dels beneficis computacionals del 2:1 balancejat (fàcil paral·lelització i alta escalabilitat), així com dels requisits mínims que aquest tipus de mallat han de complir per garantir que els espais d'elements finits que s'hi defineixin estiguin ben posats. Quant a (2), s'ha formulat un mètode robust, òptim i escalable per agregació per problemes el·líptics amb contorn o interface immerses. Després d'augmentar (1)+(2) amb un nova estratègia paral·lela per (3), el marc de simulació resultant mitiga de manera efectiva el principal coll d'ampolla en la simulació de processos de fabricació additiva en llits de pols de metall: adaptivitat i remallat escalable en geometries complexes que creixen en el temps. Durant el desenvolupament de la tesi, es col·labora amb el Monash Centre for Additive Manufacturing i la Universitat de Monash de Melbourne, Austràlia, per investigar el problema d'aplicació. En primer lloc, es fa una anàlisi experimental i numèrica exhaustiva dels mètodes d'aggregació temporal. En segon lloc, es proposa i valida experimental una nova formulació de contacte tèrmic que té en compte la inèrcia tèrmica i és adequat per a localitzar el model, l'anomenada aproximació per dominis virtuals. Mitjançant l'ús eficient de recursos computacionals d'alt rendiment, el nostre nou marc computacional fa possible l'anàlisi d'elements finits a gran escala dels processos de fabricació additiva amb metalls, amb augment de la fidelitat de les prediccions i reduccions significatives de temps de computació. Així mateix, es pot combinar amb els models reduïts que es proposen per l'optimització tèrmica del procés de fabricació. Aquestes eines contribueixen a accelerar la comprensió del lligam procés-rendiment i la digitalització del disseny i certificació de productes en fabricació additiva per metalls, dues fites crucials per explotar la tecnologia en producció en massa.Postprint (published version