Local Error Estimates Research Articles

Non‐linear space‐time elasticity

SummaryIn this contribution we introduce a novel space‐time formulation for non‐linear elasticity, able to calculate large deformations and displacements with high efficiency using structured and unstructured meshes in the space‐time cylinder without changing the required regularity and thus, enabling the use of Lagrangian shape functions including tetrahedron and hypertetrahedron or tesseract elements. The common and indiscriminate treatment of spatial and temporal directions allows us to remove one of the major bottlenecks in parallel computations: The design of time‐stepping schemes, which can neither been parallelized efficiently in time nor allow for local refinements as no information transfer backwards in time is possible. Moreover, stability of time‐stepping schemes depend on the accumulation of local approximation errors in each time‐step, in contrast to space‐time formulation, characterized by enhanced stability and robustness. Eventually, we will demonstrate superiority in the convergence against classical time‐stepping schemes using various examples including highly sensitive systems in the context of non‐linear elasticity.

An improved local error estimate in adaptive finite element analysis based on element energy projection technique

PurposeAn improved adaptive finite element analysis based on local error estimate is proposed via the element energy projection (EEP) technique. This paper aims to discuss the aforementioned idea.Design/methodology/approachThe computational region for a posteriori error estimation based on EEP method is further confined to a critical set of local elements generated in the previous adaptive step, enhancing efficiency while maintaining accuracy. The adaptive procedure incorporated with hierarchical mesh refinement is then developed.FindingsThe effectiveness of the improved error estimation of the overall adaptive analysis is confirmed by several benchmark examples. The results show that the shrinkage of the local computational region has little negative influence on the accuracy of a posteriori error estimation, thus yielding an improved adaptive procedure with simplified logic and reduced cost.Originality/valueBy localizing the computational region for error estimation, two crucial but cumbersome tricks, i.e. treatments of virtual elements and hanging nodes, are removed, giving the proposed approach full clarity and flexibility. The improved adaptive procedure characterizes simpler and faster computational algorithm and can produce results with required accuracy measured in maximum norm.

Improved local truncation schemes for the higher-order tensor renormalization group method

The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower-dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint.

Guaranteed local error estimation for finite element solutions of boundary value problems

This paper considers the finite element solution of the boundary value problem of Poisson’s equation and proposes a guaranteed local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without H2-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.

Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation

<abstract><p>We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.</p></abstract>

Performance of Inexpensive Local Error Estimation Techniques for Integral Equation Numerical Solutions

The performance of several inexpensive local error estimation techniques is evaluated in connection with the Rao-Wilton-Glisson method of moments numerical solutions of the electric field integral equation. Results for 18 perfectly conducting test targets are used to evaluate the performance of the estimators. Two of the estimators produce error maps that consistently exhibit high correlations with reference solutions. These estimators are also suitable for “goal-oriented” estimation of secondary quantities, such as identifying cells that contribute the most error to the radar cross section of the target.

Modelling edge effects at the interface in bonded joints using gradient functions in the mechanical properties of the adhesive: Application of the method to the Arcan test loaded in tension and shear

The design of bonded joints requires studies of stress concentrations due to edge effects. For complex joint configurations, the finite element method can be quite costly. The objective is to develop a fast and reliable numerical design tool for bonded assemblies. Therefore, an approach to design bonded assemblies is presented which aims to meet the needs of a design office, particularly in terms of calculation costs. The latter consists in reproducing the edge effects with a single element through the joint thickness using a modulus function. The concept was tested on Arcan specimen with two loading cases: tension (γ = 0°) and shear (γ = 90°). A 2D model under the elastic assumption is developed to describe the edge effects of the joint using Abaqus subroutines (UMat). The approach is set up to solve this problem in the form of two blocks. First, mesh refinement studies for bonded specimens loaded in tension were performed, within a good level of accuracy, on the free edge with the use of local discretization error estimation. After that, the effects of local geometry and modulus ratio are investigated. Afterwards, the von Mises stress at the interface level of the adhesive joint was used to identify a modulus function to describe the behavior of a joint with straight edge geometry for tension and shear loadings. The proposed approach has improved the performance of the model. Actually, the calculation is practically three times faster than for the conventional model.

An Optimal Adaptive Grid Method Based on L1 Scheme for a Nonlinear Caputo Fractional Differential Equation

A nonlinear fractional differential equation with a Caputo derivative of order α is studied. This problem is discretized by using the L1 scheme on an arbitrary nonuniform mesh. By utilizing the Taylor expansion with integral remainder term, an optimal local truncation error estimation of L1 scheme is proved. Based on this truncation error estimation and the mesh equidistribution principle, a new monitor function is constructed to construct an adaptive grid generation algorithm. Numerical experiments are performed to confirm the accuracy of our new adaptive grid algorithm.

Efficient reliability analysis using prediction-oriented active sparse polynomial chaos expansion

In this paper, a prediction-oriented active sparse polynomial chaos expansion (PAS-PCE) is proposed for reliability analysis. Instead of leveraging on additional techniques to reduce the problem dimensionality and/or to obtain the local error estimates, which has been done in the majority of existing PCE-based methods, this study first makes use of the Bregman-iterative greedy coordinate descent in effectively solving the least absolute shrinkage and selection operator based regression for sparse PCE approximation with a small set of initial samples. Then, the local variance distribution of the performance function is predicted using the approximated PCE. By maximizing an optimality measure that balances the exploration of design space and exploitation of the PCE characteristics, a recently proposed learning function is subsequently adopted for selecting the optimal samples one by one from a candidate pool to cover the limit state surface regions proportionally to the predicted local variance. The performance of the proposed PAS-PCE is assessed on four numerical examples of varying complexity and input dimensionality through comparison with several state-of-the-art active learning methods based on a variety of surrogate models. Results show that the proposed method is superior to the benchmark algorithms in terms of both accuracy and efficiency for reliability analysis.

New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers

Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.

Error estimates for discrete generalized FEMs with locally optimal spectral approximations

This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation spaces are constructed using eigenvectors of local eigenvalue problems solved by the finite element method on some sufficiently fine mesh with mesh size h h . The error bound of the discrete GFEM approximation is proved to converge as h → 0 h\rightarrow 0 towards that of the continuous GFEM approximation, which was shown to decay nearly exponentially in previous works. Moreover, even for fixed mesh size h h , a nearly exponential rate of convergence of the local approximation errors with respect to the dimension of the local spaces is established. An efficient and accurate method for solving the discrete eigenvalue problems is proposed by incorporating the discrete A A -harmonic constraint directly into the eigensolver. Numerical experiments are carried out to confirm the theoretical results and to demonstrate the effectiveness of the method.

Locally refined quad meshing for linear elasticity problems based on convolutional neural networks

In this paper we propose a method to generate suitably refined finite element meshes using neural networks. As a model problem we consider a linear elasticity problem on a planar domain (possibly with holes) having a polygonal boundary. We impose boundary conditions by fixing the position of a part of the boundary and applying a force on another part of the boundary. The resulting displacement and distribution of stresses depend on the geometry of the domain and on the boundary conditions. When applying a standard Galerkin discretization using quadrilateral finite elements, one usually has to perform adaptive refinement to properly resolve maxima of the stress distribution. Such an adaptive scheme requires a local error estimator and a corresponding local refinement strategy. The overall costs of such a strategy are high. We propose to reduce the costs of obtaining a suitable discretization by training a neural network whose evaluation replaces this adaptive refinement procedure. We set up a single network for a large class of possible domains and boundary conditions and not on a single domain of interest. The computational domain and boundary conditions are interpreted as images, which are suitable inputs for convolution neural networks. In our approach we use the U-net architecture and we devise training strategies by dividing the possible inputs into different categories based on their overall geometric complexity. Thus, we compare different training strategies based on varying geometric complexity. One of the advantages of the proposed approach is the interpretation of input and output as images, which do not depend on the underlying discretization scheme. Another is the generalizability and geometric flexibility. The network can be applied to previously unseen geometries, even with different topology and level of detail. Thus, training can easily be extended to other classes of geometries.

Multigrid/Multiresolution Interpolation: Reducing Oversmoothing and Other Sampling Effects

Traditional interpolation methods, such as IDW, kriging, radial basis functions, and regularized splines, are commonly used to generate digital elevation models (DEM). All of these methods have strong statistical and analytical foundations (such as the assumption of randomly distributed data points from a gaussian correlated stochastic surface); however, when data are acquired non-homogeneously (e.g., along transects) all of them show over/under-smoothing of the interpolated surface depending on local point density. As a result, actual information is lost in high point density areas (caused by over-smoothing) or artifacts appear around uneven density areas (“pimple” or “transect” effects). In this paper, we introduce a simple but robust multigrid/multiresolution interpolation (MMI) method which adapts to the spatial resolution available, being an exact interpolator where data exist and a smoothing generalizer where data are missing, but always fulfilling the statistical requirement that surface height mathematical expectation at the proper working resolution equals the mean height of the data at that same scale. The MMI is efficient enough to use K-fold cross-validation to estimate local errors. We also introduce a fractal extrapolation that simulates the elevation in data-depleted areas (rendering a visually realistic surface and also realistic error estimations). In this work, MMI is applied to reconstruct a real DEM, thus testing its accuracy and local error estimation capabilities under different sampling strategies (random points and transects). It is also applied to compute the bathymetry of Gulf of San Jorge (Argentina) from multisource data of different origins and sampling qualities. The results show visually realistic surfaces with estimated local validation errors that are within the bounds of direct DEM comparison, in the case of the simulation, and within the 10% of the bathymetric surface typical deviation in the real calculation.

NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems

In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

Local least product relative error estimation for single-index varying-coefficient multiplicative model with positive responses

In this paper, we study the single-index varying-coefficient multiplicative model (SIVCMM) for analyzing data with positive response. Based on the least product relative error (LPRE) and local kernel smoothing methods, an efficient approach is proposed for estimating the unknown parameter vector and nonparametric functions arising in SIVCMM. Furthermore, the convergence properties of the proposed estimators are established. Finally, numerical studies are conducted to test the performance of the proposed approach, and a real data analysis is presented to illustrate the new estimator’s efficiency in practical computation.

Multirate Partially Explicit Scheme for Multiscale Flow Problems

For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast. Consider the parabolic equation with heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speed. For the fast flow, we obtain a low-dimensional subspace with respect to the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.

Local Convergence of the FEM for the Integral Fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.

Local error estimate of L1 scheme for linearized time fractional KdV equation with weakly singular solutions

We consider the local error estimate of the L1 scheme on graded mesh for a linearized time fractional KdV equation with weakly singular solutions, where Legendre Petrov–Galerkin spectral method is used for the spatial discretization. Stability and convergence of the fully discrete scheme are rigorously established, and the pointwise-in-time error estimates are given to show that one can attain the optimal convergence order 2−α in positive time by mildly choosing the grading parameter r no more than 2 for all 0<α<1. Numerical results are presented to show that the error estimate is sharp.

Coverage Axis: Inner Point Selection for 3D Shape Skeletonization

AbstractIn this paper, we present a simple yet effective formulation called Coverage Axis for 3D shape skeletonization. Inspired by the set cover problem, our key idea is to cover all the surface points using as few inside medial balls as possible. This formulation inherently induces a compact and expressive approximation of the Medial Axis Transform (MAT) of a given shape. Different from previous methods that rely on local approximation error, our method allows a global consideration of the overall shape structure, leading to an efficient high‐level abstraction and superior robustness to noise. Another appealing aspect of our method is its capability to handle more generalized input such as point clouds and poor‐quality meshes. Extensive comparisons and evaluations demonstrate the remarkable effectiveness of our method for generating compact and expressive skeletal representation to approximate the MAT.

Lie-Trotter operator splitting spectral method for linear semiclassical fractional Schrödinger equation

In this paper the error estimates are derived for Lie-Trotter operator splitting spectral method for semiclassical linear fractional Schrödinger equation. We first establish a priori estimates for the solution in fractional Sobolev space. On the basis of these a priori estimates, we obtain local error bound for the well-known Lie-Trotter splitting operator associated with the linear fractional Schrödinger equation in the semiclassical regime by using a formula for the fractional Laplacian of the product of two functions. The convergence orders of the fully discrete scheme based on Fourier spectral methods for the space approximation are then analyzed and provided with respect to the time step-size Δt and the small (scaled) Planck constant ε. As far as we know, this is the first rigorous proof of local error estimate for an semi-discrete operator splitting method for semiclassical fractional Schrödinger equation. Computational experiments confirm the theoretical results.