Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff–Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.

MATHICSE Technical Report: Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

Pablo Antolin Sanchez, Annalisa Buffa, Luca Coradello, Rafael Vazquez Hernandez

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.

MATHICSE2019

Physics-informed machine learning for reduced-order modeling of nonlinear problems

Jan Sickmann Hesthaven, Qian Wang

A physics-informed machine learning framework is developed for the reduced-order modeling of parametrized steady-state partial differential equations (PDEs). During the offline stage, a reduced basis is extracted from a collection of high-fidelity solutions and a reduced-order model is then constructed by a Galerkin projection of the full-order model onto the reduced space. A feedforward neural network is used to approximate the mapping from the physical/geometrical parameters to the reduced coefficients. The network can be trained by minimizing the mean squared residual error of the reduced-order equation on a set of points in parameter space. Such a network is referred to as physics-informed neural network (PINN). As the number of residual points is unlimited, a large data set can be generated to train a PINN to approximate the reduced-order model. However, the accuracy of such a network is often limited. This is improved by using the high-fidelity solutions that are generated to extract the reduced basis. The network is then trained by minimizing the sum of the mean squared residual error of the reduced-order equation and the mean squared error between the network output and the projection coefficients of the high-fidelity solutions. For complex nonlinear problems, the projection of high-fidelity solution onto the reduced space is more accurate than the solution of the reduced-order equation. Therefore, higher accuracy than the PINN for this network - referred to as physics-reinforced neural network (PRNN) - can be expected for complex nonlinear problems. Numerical results demonstrate that the PRNN is more accurate than the PINN and both are more accurate than a purely data-driven neural network for complex problems. During the reduced basis refinement, before reaching its accuracy limit, the PRNN obtains higher accuracy than the direct reduced-order model based on a Galerkin projection.

2020

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.