Residual-based Error Estimator Research Articles

Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

Abstract The p-Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f {-\nabla\cdot((\mu+|\nabla u|^{p-2})\nabla u)=f} is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p = 3 {p=3} . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.

Goal-adaptive Meshing of Isogeometric Kirchhoff–Love Shells

AbstractMesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff–Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

An efficient and easy-to-implement recovery-based a posteriori error estimator for isogeometric analysis of the Stokes equation

In this article, we present an efficient recovery-based a posteriori error estimator for adaptive isogeometric analysis (IGA) of the Stokes equation, which is straightforward to implement. The increased regularity of splines in IGA versus Lagrange polynomials used in the classical finite element method plays a significant role for the success of the presented recovery technique. We consider LR B-splines (Johannessen et al., 2014) for the adaptive mesh refinement, and the isogeometric Taylor–Hood element, Sub-Grid element (Buffa et al., 2011), and Raviart–Thomas element (divergence-conforming) (Buffa et al., 2010; Evans and Hughes, 2013; Johannessen et al., 2015) for the mixed discretization of the Stokes equation. Three benchmark problems with analytical solutions are tested, and they demonstrate clearly the great performance of the recovery-based error estimator compared to classical residual-based error estimator.

Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

Accurate error estimation is crucial in model order reduction, to obtain small reduced-order models as well as to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.

Active-learning-driven surrogate modeling for efficient simulation of parametric nonlinear systems

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desirable. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive error-estimator-based optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider parameter-specific proper orthogonal decomposition subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks (KSNNs), abbreviated as ActLearn-POD-KSNN surrogate model. The center location for each kernel, along with center-dependent kernel widths, can be learned for the KSNN by using an alternating dual-staged iterative training procedure. To demonstrate the efficiency of our proposed ideas, we present numerical experiments using four physical models, including incompressible Navier–Stokes equations. The ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles and a fluid flow scenario with Hopf bifurcation. We also provide an investigation of the surrogate’s performance when the available data is noisy.

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a nonintrusive generalization of the adaptive Galerkin finite element method with residual-based error estimation. It combines the nonintrusive character of a randomized least squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the nonintrusive adaptive algorithm, showing the expected convergence rates of single-level strategies.

Formula omitted]-adaptive spectral projection based discontinuous Galerkin method for the numerical solution of wave equations with memory

In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (hp-adaptivity), but it is also applied to the size of the computed spectrum (k-adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum is adapted using the L2 norm of the error of the projected data. We show that the approach can be very efficient and accurate.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

Flutter Prediction Using Reduced-Order Modeling with Error Estimation

This paper presents a methodology for dynamic aeroelastic analysis of aircraft based on model order reduction with error estimation. A projection-based model order reduction approach is used to create an aerodynamic reduced-order model (ROM), which is coupled to a structural model to create an aeroelastic ROM. The governing aerodynamic equations are the linearized semidiscrete Euler equations. Flutter analysis is conducted by analyzing the eigenvalues of the aeroelastic ROM. A dual-weighted residual-based error estimator is presented which approximates the error in the eigenvalues obtained from the reduced eigenproblem relative to the eigenvalues from the high-dimensional aeroelastic model. The error estimator thus allows for the construction of aeroelastic ROMs with select eigenvalues that satisfy a user-prescribed accuracy. The aerodynamic ROM is constructed using approximate high-dimensional aeroelastic eigenvectors computed using the two-sided Jacobi–Davidson algorithm. Dynamic aeroelastic analyses are presented for the NACA 64A010 Isogai case, the AGARD 445.6 wing model, and a NACA 0012 benchmark case. The error estimator is shown to have good agreement with the exact error. For the test cases presented in this work, the cost of computing the flutter point at a given Mach number is equivalent to the cost of approximately 5 to 12 steady nonlinear flow evaluations of the high-dimensional Euler equations.

Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines

Abstract We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation and adaptivity procedure are applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated into a scan-based analysis workflow, capable of generating error-controlled results from scan data without the need for extensive user interactions or interventions.

A posteriori error estimates for finite element discretizations of time-harmonic Maxwell’s equations coupled with a non-local hydrodynamic Drude model

We consider finite element discretizations of Maxwell’s equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator drives a mesh adaptive process. These examples highlight the quality of the proposed estimator, and the potential computational savings offered by mesh adaptivity.

Unified analysis of discontinuous Galerkin andC0-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations

We provide a unified analysis ofa posteriorianda priorierror bounds for a broad class of discontinuous Galerkin andC0-IP finite element approximations of fully nonlinear second-order elliptic Hamilton–Jacobi–Bellman and Isaacs equations with Cordes coefficients. We prove the existence and uniqueness of strong solutions inH2of Isaacs equations with Cordes coefficients posed on bounded convex domains. We then show the reliability and efficiency of computable residual-based error estimators for piecewise polynomial approximations on simplicial meshes in two and three space dimensions. We introduce an abstract framework for thea priorierror analysis of a broad family of numerical methods and prove the quasi-optimality of discrete approximations under three key conditions of Lipschitz continuity, discrete consistency and strong monotonicity of the numerical method. Under these conditions, we also prove convergence of the numerical approximations in the small-mesh limit for minimal regularity solutions. We then show that the framework applies to a range of existing numerical methods from the literature, as well as some original variants. A key ingredient of our results is an original analysis of the stabilization terms. As a corollary, we also obtain a generalization of the discrete Miranda–Talenti inequality to piecewise polynomial vector fields.

Reliable Residual-Based Error Estimation for the Finite Cell Method

In this work, the reliability of a residual-based error estimator for the Finite Cell method is established. The error estimator is suitable for the application of hp-adaptive finite elements and allows for Neumann boundary conditions on curved boundaries. The reliability proof of the error estimator relies on standard arguments of residual-based a posteriori error control, but includes several modifications with respect to the error contributions associated with the volume residuals as well as the jumps across inner edges and Neumann boundary parts. Important ingredients of the proof are Stein’s extension theorem and a modified trace theorem which estimates the norm of the trace on (curved) boundary parts in terms of the local mesh size and polynomial degree. The efficiency of the error estimator is also considered by discussing an artificial example which yields an efficiency index depending on the mesh-family parameter h. Numerical experiments on more realistic domains, however, suggest global efficiency with the occurrence of a large overestimation on only few cut elements. In the experiments the reliability of the error estimator is demonstrated for h- and p-uniform as well as for hp-geometric and h-adaptive refinements driven by the error estimator. The practical applicability of the error estimator is also studied for a 3D problem with a non-smooth solution.

A residual-based artificial viscosity finite difference method for scalar conservation laws

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

Adaptive fourth-order phase field analysis using deep energy minimization

Phase field modeling of fracture is computationally expensive as it demands a very fine mesh to resolve the damage region. Hence, the practical application of such models are severely limited. Local refinement techniques are often necessary. In our recent work (Goswami et al., 2019), for solving brittle fracture problems using physics informed neural network (PINN), the crack path is resolved by minimizing the variational energy of the system. However, in Goswami et al. (2019) we used a pre-refined domain based on prior information of the failure path, which is not always available. In this work, we propose an adaptive h-refinement scheme to locally refine the domain along the path of the growth of the crack. The phase field parameter, ϕ and a residual-based posteriori error estimator are the proposed convenient measures to determine the need for refinement. For ϕ, a critical threshold is chosen such that it is lower than the value at which crack nucleation occurs and the fracture region is easily identified. On the other hand, for the residual-based error estimator, elements contributing to the highest error are marked for refinement. The proposed algorithm takes as input the geometry described using NURBS patches. For the evaluation of the basis functions, we develop a procedure based on the Bézier representation and integrate it with the adaptive refinement formulation. The results obtained using the adaptive refinement integrated variational energy based PINN approach is validated with the available analytical solution for several examples from the literature. The proposed approach is implemented on several two and three-dimensional examples to illustrate the effectiveness of the formulation. Code and data necessary for replicating the results of the examples in the article will be made available through a GitHub repository.

A hierarchical a posteriori error estimator for the Reduced Basis Method

In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of $\inf $-$\sup $ stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the $\inf $-$\sup $ constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent $\inf $-$\sup $ constant—may become infeasible.

Formula omitted]- and [formula omitted]-adaptivity driven by recovery and residual-based error estimators for PHT-splines applied to time-harmonic acoustics

In this work, we demonstrate the application of PHT-splines for time-harmonic acoustic problems, modeled by the Helmholtz equation. Solutions of the Helmholtz equation have two features: global oscillations associated with the wave number and local gradients caused by geometrical irregularities. We show that after a sufficient number of degrees of freedom is used to approximate global oscillations, adaptive refinement can capture local features of the solution. We compare residual-based and recovery-based error estimators and investigate the performance of p-refinement. The simulations are done in the context of recently introduced Geometry Independent Field approximaTion (GIFT), where PHT-splines are only used to approximate the solution, while the computational domain is parameterized with NURBS. This approach builds on the natural adaptation ability of PHT-splines and avoids the re-parameterization of the NURBS geometry during the solution refinement process.

A mixed finite element formulation for compressible finite hyperelasticity with two fibre family reinforcement

A mixed finite element formulation for compressible finite hyperelasticity, possibly with strongly deformation dependent stiffening fibre reinforcement, is developed and verified. Reinforcement by two, mechanically equivalent, in general oblique, fibre families is considered. In the provided numerical experiments, residual based error estimation is essential for a simple h-adaptive mesh refinement strategy. It is shown to be effective to resolve stress concentrations, also in combination with stress oscillations (checker boarding) in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used. The novel mixed generalised (Lagrangian) metric based formulation is illustrated using a simple but fully coupled material model. The ground substance is the common so-called compressible neo-Hooke model and the standard reinforcement part is augmented by second order anisotropic invariants. The theoretical formulation admits approaching the inextensible limit seamlessly, by-passing the use of a separate decoupled material model. The Hu–Washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables, approximated in L2 at the element level. The condensation yields a purely H1-based displacement formulation at the global level.

C0 interior penalty methods for an elliptic state-constrained optimal control problem with Neumann boundary condition

We study C0 interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains. The approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. We prove the convergence of numerical solutions with rates in the H2-like energy error by using the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm. Numerical experiments are provided to gauge the performance of the proposed methods.