Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings

Physics-aware registration based auto-encoder for convection dominated PDEs

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.Comment: 10 pages, 6 figure

Doctor of Philosophy

dissertationThe Material Point Method (MPM) and the Implicit Continuous-fluid Eulerian method (ICE) have been used to simulate and solve many challenging problems in engineering applications, especially those involving large deformations in materials and multimaterial interactions. These methods were implemented within the Uintah Computational Framework (UCF) to simulate explosions, fires, and other fluids and fluid-structure interaction. For the purpose of knowing if the simulations represent the solutions of the actual mathematical models, it is important to fully understand the accuracy of these methods. At the time this research was initiated, there were hardly any error analysis being done on these two methods, though the range of their applications was impressive. This dissertation undertakes an analysis of the errors in computational properties of MPM and ICE in the context of model problems from compressible gas dynamics which are governed by the one-dimensional Euler system. The analysis for MPM includes the analysis of errors introduced when the information is projected from particles onto the grid and when the particles cross the grid cells. The analysis for ICE includes the analysis of spatial and temporal errors in the method, which can then be used to improve the method's accuracy in both space and time. The implementation of ICE in UCF, which is referred to as Production ICE, does not perform as well as many current methods for compressible flow problems governed by the one-dimensional Euler equations - which we know because the obtained numerical solutions exhibit unphysical oscillations and discrepancies in the shock speeds. By examining different choices in the implementation of ICE in this dissertation, we propose a method to eliminate the discrepancies and suppress the nonphysical oscillations in the numerical solutions of Production ICE - this improved Production ICE method (IMPICE) is extended to solve the multidimensional Euler equations. The discussion of the IMPICE method for multidimensional compressible flow problems includes the method's detailed implementation and embedded boundary implementation. Finally, we propose a discrete adjoint-based approach to estimate the spatial and temporal errors in the numerical solutions obtained from IMPICE

Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)

This small collaborative workshop brought together experts from the Sino-German project working in the field of advanced numerical methods for hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the convergence of numerical methods and proper solution concepts were addressed as well

Three-dimensional CFD simulations with large displacement of the geometries using a connectivity-change moving mesh approach

This paper deals with three-dimensional (3D) numerical simulations involving 3D moving geometries with large displacements on unstructured meshes. Such simulations are of great value to industry, but remain very time-consuming. A robust moving mesh algorithm coupling an elasticity-like mesh deformation solution and mesh optimizations was proposed in previous works, which removes the need for global remeshing when performing large displacements. The optimizations, and in particular generalized edge/face swapping, preserve the initial quality of the mesh throughout the simulation. We propose to integrate an Arbitrary Lagrangian Eulerian compressible flow solver into this process to demonstrate its capabilities in a full CFD computation context. This solver relies on a local enforcement of the discrete geometric conservation law to preserve the order of accuracy of the time integration. The displacement of the geometries is either imposed, or driven by fluid–structure interaction (FSI). In the latter case, the six degrees of freedom approach for rigid bodies is considered. Finally, several 3D imposed-motion and FSI examples are given to validate the proposed approach, both in academic and industrial configurations

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

Adaptive Numerical Methods for PDEs

This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells