Adaptive FEM Research Articles

Rate optimal adaptive FEM with inexact solver for nonlinear operators

We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.

An adaptive FEM with ITP approach for steady Schrödinger equation

ABSTRACTIn this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank–Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.

STATIC AND DYNAMIC BEHAVIOUR ASSESSMENT OF THE TRAJAN ARCH BY MEANS OF NEW MONITORING TECHNOLOGIES

Abstract. An effective assessment of the static and dynamic structural behavior of historical monuments requires the development and validation of suitable adaptive structural models using high-quality experimental data acquired with an effectively continuous and distributed monitoring. Furthermore, the adaptive strategy allows an efficient evaluation of the health status and of the evolution along the time of a historical monument, providing relevant information to plan appropriate actions for its long-term preservation. The Trajan Arch in Benevento chosen as a case of study to develop and apply this new adaptive strategy in cultural heritage conservation. The paper, after a description of the innovative monitoring system, based on state-of-the-art mechanical sensors, presents and discusses the results of two tests, comparing the measurements with the predictions of an adaptive structural FEM model developed for the dynamical simulation of the Trajan Arch.

Adaptive unified continuum FEM modeling of a 3D FSI benchmark problem.

In this paper, we address a 3D fluid-structure interaction benchmark problem that represents important characteristics of biomedical modeling. We present a goal-oriented adaptive finite element methodology for incompressible fluid-structure interaction based on a streamline diffusion-type stabilization of the balance equations for mass and momentum for the entire continuum in the domain, which is implemented in the Unicorn/FEniCS software framework. A phase marker function and its corresponding transport equation are introduced to select the constitutive law, where the mesh tracks the discontinuous fluid-structure interface. This results in a unified simulation method for fluids and structures. We present detailed results for the benchmark problem compared with experiments, together with a mesh convergence study.

Goal‐oriented adaptivity for linear elastic micromorphic continua based on primal and adjoint consistency analysis

SummaryMicroscopic considerations are drawing increasing attention for modern simulation techniques. Micromorphic continuum theories, considering micro degrees of freedom, are usually adopted for simulation of localization effects like shear bands. The increased number of degrees of freedom clearly motivates an application of adaptive methods. In this work, the adaptive FEM is tailored for micromorphic elasticity. The proposed adaptive procedure is driven by a goal‐oriented a posteriori error estimator based on duality techniques. For efficient computation of the dual solution, a patch‐based recovery technique is proposed and compared to a reference approach. In order to theoretically ensure optimal convergence order of the proposed adaptive procedure, adjoint consistency of the FE‐discretized solution for the linear elastic micromorphic continua is shown. For illustration, numerical examples are provided. Copyright © 2017 John Wiley & Sons, Ltd.

Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form

It is shown that the h-adaptive mixed finite element method for the discretization of eigenvalue clusters of the Laplace operator produces optimal convergence rates in terms of nonlinear approximation classes. The results are valid for the typical mixed spaces of Raviart-Thomas or Brezzi-Douglas-Marini type with arbitrary fixed polynomial degree in two and three space dimensions.

Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Gårding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated.

Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM

Only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes. The BEM-based Finite Element Method is one of the new discretization strategies, which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally. The article in hand addresses quasi-interpolation operators for the approximation space over polygonal meshes. To prove interpolation estimates the Poincaré constant is bounded uniformly for patches of star-shaped elements. These results give rise to the residual based error estimate for high order BEM-based FEM and its reliability as well as its efficiency are proven. Such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive FEM strategies. Numerical experiments show optimal rates of convergence for meshes with non-convex elements on uniformly as well as on adaptively refined meshes.

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated.

Adaptive FEM and a posteriori error control for stationary coupled bulk‐surface PDEs

AbstractWe consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. The numerical error of the finite element solution can be controlled by a residual a posteriori error estimator which takes into account the approximation errors due to the discretisation in space as well as the polyhedral approximation of the surface. The estimators naturally lead to refinement indicators for an adaptive algorithm to control the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Full-discrete adaptive FEM for quasi-parabolic integro-differential PDE-constrained optimal control problem

In this paper, we develop a full-discrete adaptive FEM for a quasi-parabolic integro-differential PDE-constrained optimal control problem. Firstly, we present weak formulations and optimal conditions and, by using the backward Euler scheme, we deduce equivalent a posteriori error estimators. Then we derive lower and upper bounds for the error estimates by bubble function techniques and use these indicators in adaptive finite element schemes. Furthermore, we carry out some numerical experiments to test our theoretical results.

An adaptive least-squares FEM for the Stokes equations with optimal convergence rates

This paper introduces the first adaptive least-squares finite element method (LS-FEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowest-order Raviart-Thomas and conforming $$P_1$$P1 discrete spaces for the divergence least-squares formulation in 2D. Although the least-squares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residual-based a posteriori error control with exact solve. Particular interest is on the treatment of the data approximation error which requires a separate marking strategy. The paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non-linear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:43---62 2015) from the Poisson model problem to the Stokes equations.

Adaptive mesh refinement FEM for seismic damage evolution in concrete-based structures

In order to study the seismic damage mechanisms of concrete-based structures, a adaptive multi-grid method is developed to simulate the process from evolving damage to structural failure in concrete structures under seismic loading. The theory of multi-grid FEM coupled evolving damage is developed on the basis of the improved variational principle, in which the elements in each sub-domain with different grid sizes are under the different state of damage. Meanwhile the FE elements can be refined adaptively with the accumulation of its damage for lower cost, without user intervention in the computation. As a case study of the method, the failure process of reinforced concrete column under seismic loading is simulated, the results show that, the simulated results fit well with the experiment, and the developed method can be used to reveal the seismic damage mechanism of concrete-based structures by considering the process from material damage to structural failure; the adaptive multi-grid method can give satisfactory result with lower cost by compared with experiment.

One-dimensional fully automatic h-adaptive isogeometric finite element method package

This paper deals with an adaptive finite element method originally developed by Prof. Leszek Demkowicz for hierarchical basis functions. In this paper, we investigate the extension of the adaptive algorithm for isogeometric analysis performed with B-spline basis functions. We restrict ourselves to h-adaptivity, since the polynomial order of approximation must be fixed in the isogeometric case. The classical variant of the adaptive FEM algorithm, as delivered by the group of Prof. Demkowicz, is based on a two-grid paradigm, with coarse and fine grids (the latter utilized as a reference solution). The problem is solved independently over a coarse mesh and a fine mesh. The fine-mesh solution is then utilized as a reference to estimate the relative error of the coarse-mesh solution and to decide which elements to refine. Prof. Demkowicz uses hierarchical basis functions, which (though locally providing C p−1 continuity) ensure only C 0 on the interfaces between elements. The CUDA C library described in this paper switches the basis to B-spline functions and proposes a one-dimensional isogeometric version of the h-adaptive FEM algorithm to achieve global C p−1 continuity of the solution.

Preliminary Upper Estimate of Peak Currents in Transcranial Magnetic Stimulation at Distant Locations From a TMS Coil.

Transcranial magnetic stimulation (TMS) is increasingly used as a diagnostic and therapeutic tool for numerous neuropsychiatric disorders. The use of TMS might cause whole-body exposure to undesired induced currents in patients and TMS operators. The aim of this study is to test and justify a simple analytical model known previously, which may be helpful as an upper estimate of eddy current density at a particular distant observation point for any body composition and any coil setup. We compare the analytical solution with comprehensive adaptive mesh refinement-based FEM simulations of a detailed full-body human model, two coil types, five coil positions, about 100000 observation points, and two distinct pulse rise times; thus, providing a representative number of different datasets for comparison, while also using other numerical data. Our simulations reveal that, after a certain modification, the analytical model provides an upper estimate for the eddy current density at any location within the body. In particular, it overestimates the peak eddy currents at distant locations from a TMS coil by a factor of 10 on average. The simple analytical model tested in this study may be valuable as a rapid method to safely estimate levels of TMS currents at different locations within a human body. At present, safe limits of general exposure to TMS electric and magnetic fields are an open subject, including fetal exposure for pregnant women.

Dual‐based adaptive FEM for inelastic problems with standard FE implementations

SummaryGoal‐oriented error estimation allows to refine meshes in space and time with respect to arbitrary quantities. The required dual problems that need to be solved usually require weak formulations and the Galerkin method in space and time to be established. Unfortunately, this does not obviously leads to structures of standard finite element implementations for solid mechanics. These are characterized by a combination of variables at nodes (e.g. displacements) and at integration points (e.g. internal variables) and are solved with a two‐level Newton method because of local uncoupled and global coupled equations. Therefore, we propose an approach to approximate the dual problem while maintaining these structures. The primal and the dual problems are derived from a multifield formulation. Discretization in time and space with appropriate shape functions and rearrangement yields the desired result. Details on practical implementation as well as applications to elasto‐plasticity are given. Numerical examples demonstrate the effectiveness of the procedure. Copyright © 2015 John Wiley & Sons, Ltd.

Computation of aeroacoustic sources for a Gulfstream G550 nose landing gear model using adaptive FEM

This work presents a direct comparison of unsteady, turbulent flow simulations with measurements performed using a Gulfstream G550 nose landing gear model. The experimental campaign, which was carried out by researchers from the NASA Langley Research Center, provided a series of detailed, well documented wind-tunnel measurements for comparison and validation of computational fluid dynamics (CFD) and computational aeroacoustics (CAA) methodologies. Several computational efforts were collected and presented at the Benchmark for Airframe Noise Computation workshops, BANC-I and II. For our simulations, we used a General Galerkin finite element method (G2), where no explicit subgrid model is used, and where the computational mesh is adaptively refined with respect to a posteriori estimates of the error in a quantity of interest, here the source term in Lighthill’s equation. The mesh is fully unstructured and the solution is time-resolved, which are key ingredients for solving problems of industrial relevance in the field of aeroacoustics. Moreover, we choose to model the boundary layers on the landing gear geometry with a free-slip condition for the velocity, which we previously observed to produce good results for external flows at high Reynolds numbers, and which considerably reduces the amount of cells required in the mesh. The comparisons presented here are an attempt to quantify the accuracy of our models, methods and assumptions; to that end, several results containing both time-averaged and unsteady flow quantities, always side by side with corresponding experimental values, are reported. The main finding is that we are able to simulate a complex, unsteady flow problem using a parameter-free methodology developed for high Reynolds numbers, external aerodynamics and aeroacoustics applications.

Adaptive FEM with goal‐oriented error estimation and an approximation of the dual problem for inelastic problems

AbstractAn approximation of the dual problem for goal‐oriented error estimation is derived, which preserves the numerical structure of the primal problem, usually found in standard finite element implementations for solid mechanics. For the error estimation we apply a higher‐order recovery procedure. An application to elasto‐plasticity is given. Numerical examples demonstrate the effectiveness of the procedure. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Biorthogonal basis functions in [formula omitted]-adaptive FEM for elliptic obstacle problems

In this paper, the discretization of a non-symmetric elliptic obstacle problem with hp-adaptive H1(Ω)-conforming finite elements is discussed. For this purpose, a higher-order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions. The hp-adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper. The use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi-smooth Newton scheme, which can be modified to ensure global convergence. hp-adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the H1(Ω)-conforming finite element discretization. This is typically done by using so-called connectivity matrices. In this paper, a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions. Finally, the applicability of the theoretical findings is demonstrated with several numerical experiments.

Adaptive FEM with Domain Decomposition Method for Partitioned-Based Fluid–Structure Interaction

A numerical assessment on the solution performance of adaptive finite element methods for fluid–structure interaction (FSI) is presented in this paper. The partitioned-based approach involving separate fluid and structure solvers is considered for adaptive finite element computation with triangular mesh. The performance of the hp-adaptivity, in which error in energy norm is reduced by way of mesh refinement (h) and polynomial order extension (p), is of particular interest. In addition, parallel solution process based on the domain decomposition method is also assessed due to the complexity of two way coupling scheme. Based on its numerical convergence, the hp-adaptive procedure for the FSI problem is shown to yield the fastest convergence as generally expected. Moreover, our domain decomposition parallelization scheme shows reduction in the computation time by up to order 2. Subsequently, the convergence performance is highly dependent on the aspect ratios of the triangular elements. The solution performance strongly suggests the viability of the parallel hp-adaptive method in partitioned-based FSI formulation.