Adaptive Finite Element Research Articles

Error assessment of an adaptive finite elements—neural networks method for an elliptic parametric PDE

We present a finite elements—neural network approach for the numerical approximation of parametric partial differential equations. The algorithm generates training data from finite element simulations, and uses a data-driven (supervised) feedforward neural network for the online approximation of the solution. The objective is to ensure that the overall error of the method is below some preset tolerance, and we thus control and balance the error coming from the finite element method, and the one introduced by the neural network approximation.Two finite element methods are considered and compared; a fixed grid approach uses the same mesh for all values of the parameters, while an adaptive finite elements approach enforces the same discretization error uniformly in the parameters space.Numerical results are presented for an elliptic model problem. The fixed grid approach shows limitations in terms of error balancing and control, while the adaptive approach allows a better accuracy and more flexibility of the method. We conclude by proposing an adaptive algorithm to control the size of the training set given a network architecture and ensure that the overall error of the method is below a given tolerance.

Centrifuge and analytical modeling of counterweight retaining walls under translation mode

Counterweight retaining walls (CRWs) are a variant of gravity-retaining walls that feature a pressure relief platform (RP) on the backfill side to reduce total earth pressure, offering a cost-effective option compared to traditional forms. However, the intricate interactions between CRWs and backfill, coupled with the lack of clear design guidelines, have restricted their use. This study examines the behavior of CRWs in translation mode using three different centrifuge model tests, each with varying fill heights. Observations show a slightly concave slip surface through the wall heel and a nearly planar slip surface extending from the edge of the RP to the back of the wall in an active limit state. Backfill movement can be classified into translation, sliding, and stable zones. Earth pressure distribution on the upper wall follows a trapezoidal pattern, whereas the pressure on the lower wall exhibits a triangular distribution. Earth pressure on the RP is approximately linear, peaking near the RP’s edge. Introducing a friction angle mobilization factor for the potential first slip surface improves accuracy in calculating earth pressure. The tests were replicated to validate the analytical model for earth pressure calculations, using the adaptive finite-element limit analysis method.

Adaptive meshless local maximum-entropy finite element method for Navier-Stokes equations

Abstract Based on the successful application of the adaptive meshless local maximum-entropy finite element method to solve the convection-diffusion equation, this study extends the same principle to study the 2-dimensional Navier-Stokes equations. Through extensive case studies, this work demonstrates that the present approach is a viable alternative to resolve the high Reynolds number Navier-Stokes equations. The simulation results indicate that by incorporating additional points into the elements without increasing the bandwidth or refinement via the local maximum-entropy procedure, it will enhance the accuracy and efficiency of numerical simulations. A 2-dimensional square lid-driven cavity with various Reynolds numbers will serve as the first example. In the second example, we address a more complex geometry by solving the cavity with a hole inside the cavity center. The numerical results of the model compare favorably with other numerical solutions, including the finite difference method and the finite element method. This paper provides a very powerful tool to study the boundary layer theory with irregular geometry of the Navier-Stokes equations.

Mesh Compression Algorithm for Geometrical Coordinates in Computational Meshes

Application of advanced mesh based methods, including adaptive finite element method, is impossible without theoretical elaboration and practical realization of a model for organization and functionality of computational mesh. One of the most basic mesh functionality is storing and providing geometrical coordinates for vertices and other mesh entities. New algorithm for this task based on on-the-fly recreation of coordinates was developed. Conducted tests are proving that, for selected cases, it can be orders of magnitude faster than naive approach or other similar algorithms.

Quasi-Optimality of an AFEM for General Second Order Elliptic PDE

Abstract In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ⁢ ( Ω ) ] d {b\in[L^{\infty}(\Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 {H^{1}} -regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.

An Efficient Goal-Oriented Adaptive Finite Element Method for Accurate Simulation of Complex Electromagnetic Radiation Problems

A goal-oriented adaptive frequency domain finite element method for solving electromagnetic radiation problems including complex structures is presented in this paper. Compared with the traditional adaptive finite element method, the goal-oriented method can flexibly control the refined regions according to the parameters of interest; therefore it has better convergence and has made significant progress in scattering problems and eigenvalue problems. To simulate complex antennas, this paper proposes an error indicator with high accuracy and low computational cost, and it uses the adjoint problem to weight element residuals without additional degrees of freedom. Moreover, high-quality mesh refinement algorithms adapted to this indicator are developed using a suitable point insertion strategy for multiscale structures. By simulating two practical antennas, comparisons with the traditional goal-oriented FEM and the well-developed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -adaptive finite element method in commercial software demonstrate the accuracy and efficiency of the proposed method.

3-D Forward Modeling of Semi-Airborne TEM Method With IP Effect Based on Goal-Oriented Adaptive Finite Element Algorithm

3-D Forward Modeling of Semi-Airborne TEM Method With IP Effect Based on Goal-Oriented Adaptive Finite Element Algorithm

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Local multigrid method for solving adaptive finite element equations of three-dimensional elasticity problems

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

3-D Joint Inversion of MT and CSEM Data Based on Adaptive Finite-Element Forward Modeling

3-D Joint Inversion of MT and CSEM Data Based on Adaptive Finite-Element Forward Modeling

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

An Efficient Parallel Iterative Solver for Controlled-source Electromagnetic 3-D Adaptive Forward Modeling in General Anisotropic Media

Controlled-source electromagnetic (CSEM) surveys are moving increasingly toward realistic and complicated scenarios where the survey region may contain undulated topography, complex geometries, and electrical anisotropy media. In this paper, the adaptive finite element numerical method is employed to discrete the total electric field equation, which can provide precise electromagnetic (EM) responses even with a coarse initial mesh. The unstructured tetrahedral grid is employed to effectively address arbitrary irregular geometry and mountain terrain. Then, the flexible generalized minimum residual solver (FGMRES), auxiliary-space Maxwell pre-conditioner, and grid division technique were used to solve the large-scale linear system of equations, which can stably solve ill-conditioned problems using fewer computing resources. Finally, we conducted a numerical experiment via our newly proposed forward modeling scheme on a synthetic model with multi-contrast electrical anisotropy. It validated that accurate EM fields could be obtained against the semi-analytic solutions, and this iterative solver has good robustness for various anisotropic media. As a result, we have developed a state-of-the-art 3D CSEM anisotropic forward modeling engine, which can quickly and accurately deal with large-scale and complex geo-models.

Gaussian process regression-driven deep drawing blank design method

This research introduces a machine learning (ML)-based methodology for the optimal blank design of components manufactured through the deep drawing process, considering the interplay among material, process, and geometric parameters. The proposed blank design mapping function (BDMF) leverages a Gaussian process regression (GPR) ML model in conjunction with a radial basis function (RBF) kernel. This combination allows for correlating predictions with their standard deviations, capturing the estimations' quality effectively. The GPR model was trained using the results from a three-dimensional adaptive mesh-based finite element analysis (FEA) model, characterized by a fixed node count of 23 input parameters, an explicit solution scheme, and an average computational time of 270 s. Laboratory-scale experiments on an R47.5 mm flanged cup constructed from AISI-304 steel and AA5754 aluminum alloy served to validate the FEA models and the proposed BDMF. The comparison between experimental outcomes and FEA results revealed maximum deviations of 13.3 % in the drawing force and 0.35 % for the earing profile over a 90° segment. The comparison between experimental data and BDMF predictions for the sheet metal blank indicated average deviations of 0.015 mm (or 1.3 %) in estimating thickness and 0.12 mm (or 0.25 %) in predicting the outer radius. Application of the BDMF to four additional flanged geometries with varying shapes demonstrated its reliability and generality; the maximum and average deviations for the earing profile were 4.3 % and 3.1 %, respectively, and for post-forming sheet thickness, they were 6.7 % and 3.1 %, respectively.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

Equiaxed growth of interacting Al–Cu dendrites in thin samples: a phase-field study at copper concentrations relevant for practical applications

We perform three-dimensional phase-field simulations of equiaxed solidification in Al–Cu thin samples. Purely diffusive conditions are considered in order to describe systems where convection and gravity effects can be neglected. The use of a parallel adaptive finite element algorithm introduced recently [Gong et al., Comput. Mater. Sci. 147 (2018) p. 338-352] allows us to reach the domain of copper concentrations used in practical applications (c≥3 wt% Cu). We compare the present results with those of a previous study which was restricted to lower copper concentrations (c≤2 wt% Cu) [Boukellal et al., Materialia 1 (2018) p. 62-69] due to the use of a finite difference code. In the fast dendritic growth regime, our results confirm that the dimensionless growth length Λ is independent of the copper concentration and the average separation distance between the dendrite nuclei. The new data obtained at higher copper concentrations lead to a more accurate estimate of Λ. Physical arguments are developed to specify the meaning of Λ and the grounds of the scaling law Λ=cst. Comparisons with available experimental results of the literature give additional support to this scaling law.

Discrete Approximation of the Griffith Functional by Adaptive Finite Elements

Discrete Approximation of the Griffith Functional by Adaptive Finite Elements

On the construction of an efficient finite-element solver for phase-field simulations of many-particle solid-state-sintering processes

We present an efficient solver for the simulation of many-particle solid-state-sintering processes. The microstructure evolution is described by a system of equations consisting of one Cahn–Hilliard equation and a set of Allen-Cahn equations to distinguish neighboring particles. The particle packing is discretized in space via multicomponent linear adaptive finite elements and implicitly in time with variable time-step sizes, resulting in a large nonlinear system of equations with strong coupling between all components to be solved. Since on average 10k degrees of freedom per particle are necessary to accurately capture the interface dynamics in 3D, we propose strategies to solve the resulting large and challenging systems. This includes the efficient evaluation of the Jacobian matrix as well as the implementation of Jacobian-free methods by applying state-of-the-art matrix-free algorithms for high and dynamic numbers of components, advances regarding preconditioning, and a fully distributed grain-tracking algorithm. We validate the obtained results, examine in detail the node-level performance and demonstrate the scalability up to 10k particles on modern supercomputers. Such numbers of particles are sufficient to simulate the sintering process in (statistically meaningful) representative volume elements. Our framework thus forms a valuable tool for the virtual design of solid-state-sintering processes for pure metals and their alloys.

The adaptive SAV weak Galerkin finite element method for the Allen-Cahn equation

In this paper, the weak Galerkin finite element method with the scalar auxiliary variable (SAV) approach is considered for the Allen-Cahn equation. Based on the elliptic reconstruction technique, the elliptic equation corresponding to the Allen-Cahn equation is introduced, which is employed to split the numerical error into the elliptic error and the parabolic error. Then the weak gradient recovery type a posteriori error estimator of the elliptic equation is adopted to develop the time-space adaptive algorithm. The effectiveness of the SAV weak Galerkin finite element method and the time-space adaptive algorithm is verified by several numerical benchmarks on both uniform and adaptive meshes.

Solver algorithm for stabilized space-time formulation of advection-dominated diffusion problem

This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution using higher-order continuous B-spline basis functions in both spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.

Field-based recovery technique for improved adaptive finite element analysis of photonic devices

A new adaptive finite element (FE) technique is proposed to enhance the accuracy of the solution of wave propagation in integrated photonic devices. The suggested recovery technique improves significantly the finite element convergence rate compared to refinement techniques. The method refines the elements based on the field distribution. This recovery technique estimates the error at each element by comparing the linear FE solution and the recovery solution using least-squares method. The numerical aspects and accuracy of the method are investigated using the light propagation in both one-dimensional and two-dimensional photonic devices.