Automatic Mesh Refinement Research Articles

A symmetry and Noether charge preserving discretization of initial value problems

Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.

Capturing photoionization shadows in streamer simulations using the discrete ordinates method

Numerical simulations of streamer propagation involving photoionization are presented, utilizing an ANSYS Fluent implementation that employs unstructured meshes and automatic mesh refinement. Two approximate methods for radiative transfer, used to handle computation of the photoionization source terms, are compared: the Eddington approximation and the discrete ordinates (DOs) method. The former is commonly employed in streamer simulations, while the latter is well-established in other branches of computational physics, such as radiative heat transfer. A 2D test case with two distinct regions, where streamer propagation can be triggered thanks to the protruded electrodes, is introduced. The two regions are partially separated by an opaque solid insulator barrier to study the effects of photoionization shadows on streamer inception and propagation. The primary positive streamer is initiated by placing a neutral plasma patch close to one of the electrode protrusions, while the secondary positive streamer, in the other region of the computational domain, is initiated by photoionization originating from the primary streamer zone. The Eddington approximation results in an excessively high photoionization source in the secondary streamer inception zone, as it fails to capture the shadowing effects of the opaque dielectric barrier. Consequently, this leads to a fast secondary streamer inception process, followed by rapid streamer propagation. On the other hand, the DOs method accurately captures the shadow, leading to a delayed secondary streamer inception. It is also shown that both methods exhibit very similar results when the dielectric barrier is transparent and the shadow is absent. This work demonstrates that using the DOs method for streamer simulations offers considerable advantages over the Eddington approximation, especially in cases involving more complex geometries where shadows need to be captured for accurate streamer inception and dynamics.

Error-driven adaptive mesh refinement for unsteady turbulent flows in spectral-element simulations

The simulation of turbulent flows requires high spatial resolution in potentially a priori unknown, solution-dependent locations. To achieve adaptive refinement of the mesh, we rely on error indicators. We assess the differences between an error measure relying on the local convergence properties of the numerical solution and a goal-oriented error measure based on the computation of an adjoint problem. The latter method aims at optimizing the mesh for the calculation of a predefined integral quantity, or functional of interest. This work follows on from a previous study conducted on steady flows in Offermans et al. (2020) and we extend the use of the so-called adjoint error estimator to three-dimensional, turbulent flows. They both represent a way to achieve error control and automatic mesh refinement (AMR) for the numerical approximation of the Navier–Stokes equations, with a spectral element method discretization and non-conforming h-refinement.The current study consists of running the same physical flow case on gradually finer meshes, starting from a coarse initial grid, and to compare the results and mesh refinement patterns when using both error measures. As a flow case, we consider the turbulent flow in a constricted, periodic channel, also known as the periodic hill flow, at four different Reynolds numbers: Re = 700, Re = 1400, Re = 2800 and Re = 5600. Our results show that both error measures allow for effective control of the error, but they adjust the mesh differently. Well-resolved simulations are achieved by automatically focusing refinement on the most critical regions of the domain, while significant saving in the overall number of elements is attained, compared to statically generated meshes. At all Reynolds numbers, we show that relevant physical quantities, such as mean velocity profiles and reattachment/separation points, converge well to reference literature data. At the highest Reynolds number achieved (Re = 5600), relevant quantities, i.e. reattachment and separation locations, are estimated with the same level of accuracy as the reference data while only using one-third of the degrees of freedom of the reference. Moreover, we observe distinct mesh refinement patterns for both error measures. With the spectral error indicator, the mesh resolution is more uniform and turbulent structures are more resolved within the whole domain. On the other hand, the adjoint error estimator tends to focus the refinement within a localized zone in the domain, dependent on the functional of interest, leaving large parts of the domain marginally resolved.

Fully automatic multigrid adaptive mesh refinement strategy with controlled accuracy for nonlinear quasi-static problems

We propose an adaptive mesh refinement (AMR) algorithm dedicated to the simulation of nonlinear quasi-static solid mechanics problems with complex local phenomena at the structural scale. The proposed method allows us to follow in time the evolution of studied phenomena in a fully-automatic (based on error estimators), precise (respecting user-prescribed accuracies) and efficient (in terms of memory space and computational time) way. This algorithm is based on the multilevel Local Defect Correction (LDC) refinement approach. We first introduce an algorithmic extension of the LDC method to nonlinear quasi-static problems and provide key aspects associated with its practical implementation. Generic still open AMR-related questions associated with dynamic mesh adaptation, such as fields transfer between time steps and discretization error control over time, are then addressed. We propose a straightforward and efficient error non accumulation strategy lying on the introduction of the unbalance residual as an initial source term of the problem. Moreover, a reliable remeshing algorithm is introduced, aiming to limit the number of mesh regenerations over time while guaranteeing the fulfillment of user-prescribed errors. The efficiency of the proposed algorithm is demonstrated on several numerical experiments, in 2D and 3D, with different types of material behavior as well as evolving loads. Thanks to its natural ability to generate a hierarchy of meshes of limited sizes that dynamically follow the evolution over time of studied phenomena, the proposed extension of the LDC method clearly appears to be of great potential for many challenging applications.

An efficient automatic adaptive algorithm for cohesive crack propagation modeling of concrete structures using matrix-free unstructured Galerkin Finite Volume Method

Using Galerkin Finite Volume method, a 2D adaptive algorithm is developed for crack propagation simulation of concrete structures under mixed mode loading. The matrix free Finite Volume formulation is applied to implement a crack propagation model in a fully automatic adaptive mesh refinement framework for unstructured meshes. Cohesive cracks are implemented using nonlinear interface elements with bilinear softening constitutive equation. On contrary to existing cohesive crack models where the Newton-Raphson techniques are required to simulate the nonlinear behavior in the fracture process zone, the need for such a computational costly technique is relaxed. The matrix-free Finite Volume formulation method can provide better computational performance than the matrix-base methods. The introduced Method can also relax the problems of the standard Finite Element base methods with high aspect-ratio elements. The present Finite Volume model accurately predicts the crack propagation path and correctly evaluates the load-displacement curve while consumes low computational CPU time.

Phase change driven adaptive mesh refinement in PFEM

The particle finite element method (PFEM) is used to simulate a simple phase change problem. This is a first step towards the simulation of additive manufacturing (AM) processes at the meso-scale, where the liquid melt pool interacts with the surrounding solid material and undergoes phase change. The focus of this paper lies on strategies to deal with the release or absorption of latent heat in the PFEM, especially with regard to mesh refinement. We briefly describe how mesh refinement in PFEM works and how it can be chosen specifically to achieve convergence despite the highly non-linear latent heat term. It is found that good agreement with the literature can be achieved on a simple 1D phase change test case, while using an automatic local mesh refinement.

Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements.

The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

An adaptive multigrid solver for DPG methods with applications in linear acoustics and electromagnetics

We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov–Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. The key point of the construction is the integration of the iterative solver with a fully automatic and reliable mesh refinement process provided by the DPG technology. The efficacy of the solution technique is showcased with numerous examples of linear acoustics and electromagnetic simulations, including simulations in the high-frequency regime, problems which otherwise would be intractable. Finally, we analyze the one-level preconditioner (smoother) for uniform meshes and we demonstrate that theoretical estimates of the condition number of the preconditioned linear system can be derived based on well established theory for self-adjoint positive definite operators.

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection–diffusion–reaction problems.

Robust adaptive topology optimization of porous infills under loading uncertainties

The majority of topology optimization methods for porous infill designs is based on the assumption of deterministic loads. However, in practice, quantities such as positions, weights, and directions of applied loads may change accidentally. Deterministic load-based designs might deliver poor structural performance under loading uncertainties. Such uncertain factors need to be taken into account in topological optimization to seek robust results. This paper presents a novel robust concurrent topology optimization method for the design of uniform/non-uniform porous infills under the accidental change of loads. A combination of moving morphable bars (MMBs) and loading uncertainties is proposed to directly model multiscale structures and seek robust designs. The macro- and microscopic structures can be simultaneously optimized through the minimization of the weighted sum of the expected compliance and standard deviation. The geometries of adaptive geometric components (AGCs) are straightforwardly optimized. The AGCs consist of two classes of geometric components: macroscopic bars describing the overall structure and microscopic bars describing the material microstructures. Automatic mesh-refinement is utilized to enhance computing efficiency. Numerical examples demonstrate that robust porous design can be obtained with only one global volume constraint while the material continuity of neighboring unit cells and the structural porosity can be maintained without additional constraints. The robust designs yield a more robust structural performance along with a smaller standard deviation compared with deterministic porous designs under loading uncertainties.

Increasing force generation in electroadhesive devices through modelling of novel electrode geometries

Two approaches were taken to increase the obtainable electroadhesive (EA) forces from EA electrode devices using finite element analysis (FEA): (i) optimising electrode widths and spacings using 2D parametric simulations; and (ii) optimising electrode geometries using 3D simulations. The Maxwell stress distributions generated by the simulated EA pads are reported; and illustrate how electrode geometries influence the EA forces generated. The FEA analyses were carried out using ANSYS MAXWELL with an automatic adaptive mesh refinement (AMR) technique which accelerates convergence and decreases the number of elements needed for study of mesh independency. The 2D parametric FEA shows optimum electrode widths of 2.6 mm and optimum spacing between electrodes of 0.2 mm for an effective surface of EA pads. 3D FEA shows that from the studied EA pads with a constant effective area, sine-wave shaped electrodes can generate substantially enhanced EA forces compared to other electrode geometries. This is attributed to maximising the electric field gradient and could benefit multiple applications. The simulation procedure is also validated by real-world problems.

An efficient stress recovery technique in adaptive finite element method using artificial neural network

In this paper, an efficient stress recovery technique is presented to estimate the recovered stress field at the nodal points. The feed–forward back–propagation multilayer perceptron (MLP) neural network approach is employed to improve the accuracy of the stress recovery method. An automatic adaptive mesh refinement is performed based on a–posteriori Zienkiewicz–Zhu error estimation method. The proposed technique is employed to recover the stress field accurately in the regions with a high stress gradient where the conventional recovery techniques are not able to improve the stress fields efficiently due to the singular behavior of problem. Finally, several numerical examples are solved to demonstrate the efficiency and accuracy of the proposed computational algorithm. The results are compared with the conventional methods, including the averaging method, superconvergent patch recovery (SPR) technique, and weighted superconvergent patch recovery (WSPR) method that illustrates how the artificial neural network can be used accurately to recover the stress field.

Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains

We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modeling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.

A new adaptive mesh refinement strategy based on a probabilistic error estimation

In this paper, an automatic adaptive mesh refinement procedure is presented for two-dimensional problems on the basis of a new probabilistic error estimator. First-order perturbation theory is employed to determine the lower and upper bounds of the structural displacements and stresses considering uncertainties in geometric sizes, material properties and loading conditions. A new probabilistic error estimator is proposed to reduce the mesh dependency of the responses dispersion. The suggested error estimator neglects the refinement at the critical points with stress concentration. Therefore, the proposed strategy is combined with the classic adaptive mesh refinement to achieve an optimal mesh refined properly in regions with either high gradients or high dispersion of the responses. Several numerical examples are illustrated to demonstrate the efficiency, accuracy and robustness of the proposed computational algorithm and the results are compared with the classic adaptive mesh refinement strategy described in the literature.

Polygonal finite element modeling of crack propagation via automatic adaptive mesh refinement

Polygonal finite element provides a great flexibility in mesh generation of crack propagation problems where the topology of the domain changes significantly. However, the control of the discretization error in such problems is a main concern. In this paper, a polygonal-FEM is presented in modeling of crack propagation problems via an automatic adaptive mesh refinement procedure. The adaptive mesh refinement is accomplished based on the Zienkiewicz–Zhu error estimator in conjunction with a weighted SPR technique. Adaptive mesh refinement is employed in some steps for reduction of the discretization error and not for tracking the crack. In the steps that no adaptive mesh refinement is required, local modifications are applied on the mesh to prevent poor polygonal element shapes. Finally, several numerical examples are analyzed to demonstrate the efficiency, accuracy and robustness of the proposed computational algorithm in crack propagation problems.

Adaptive Finite Element Method of Lines with Local Mesh Refinement in Maximum Norm Based on Element Energy Projection Method

The reliable and efficient self-adaptive analysis is a modern goal of various numerical computations. Most adaptivity methods, however, adopt energy norm to measure errors, which may not be the most natural and convenient means, e.g., for problems with locally singular gradient of displacement. Based on the Element Energy Projection (EEP) super-convergent technique in the Finite Element Method of Lines (FEMOL) which is a general and powerful semi-discrete method, reliable error estimates of displacements in maximum norm can be obtained anywhere on the FEMOL mesh and hence adaptive FEMOL by maximum norm becomes feasible. However, to tackle singularity problems effectively and efficiently, an automatic and flexible local mesh refinement strategy is required to generate meshes of high quality for more efficient adaptive FEMOL analysis. Taking the two-dimensional Poisson equation as the model problem, the paper firstly introduces the FEMOL and EEP methods with interface sides resulting from local mesh refinement. Then a local mesh refinement strategy and corresponding adaptive algorithm are presented. The numerical results given show that the proposed adaptive FEMOL with local mesh refinement can produce displacement solutions satisfying the specified tolerances in maximum norm and the adaptively-generated meshes reasonably reflect the local difficulties inherent in the physical problems without much redundant accuracy.

A Mesh Refinement Algorithm for Singularly Perturbed Boundary and Interior Layer Problems

A robust automated mesh refinement algorithm is proposed for numerical solution of singularly perturbed boundary and interior layer problems. The proposed algorithm is heuristically based on two smoothness indicators which are functions of consecutive differences of the computed solution on a given mesh. The main salient features of the proposed heuristic mesh generation algorithm are: (i) it does not require a priori knowledge about the nature or location of the layered region, (ii) it is independent of the problem under consideration and the underlying numerical scheme. The proposed algorithm is applied to solve a wide range of 1D as well as 2D test problems to show its robustness.

A 3D automatic mesh refinement X-FEM approach for fatigue crack propagation

A new approach combining the eXtended Finite Element Method (X-FEM) and automatic Adaptive Mesh Refinement (AMR) is presented taking advantage of both methods. The X-FEM, developed over the past two decades by a large community, have proven its efficiency to handle evolving discontinuities in a variety of fracture analysis. Since this method enables to describe the crack and its propagation independently of the mesh of the structure, a simple hierarchical mesh refinement procedure can be applied. Automatic adaptive remeshing is a valuable method in 3D elastic-plastic crack propagation analysis since it permits a locally fine mesh and then an accurate description of physical quantities in a limited area around the crack front. This is particularly important when local fracture criteria are concerned. Moreover local refinement saves computational effort, which is relevant for engineer-sized problems, particularly when the propagation path is not a priori known. In the present work, it is shown that both methods combine with minimal effort: the kinematic continuity relations and the field transfer process, needed for history-dependent material, must include in a proper way the enrichment of the model. If this requirement is not fulfilled, numerical error may be introduced. Implementation of this combined X-FEM/AMR approach is presented in detail. In particular, an innovative field transfer strategy is proposed. Numerical applications in 2 and 3 dimensions of crack propagation in elastic-plastic media demonstrate accuracy, robustness and efficiency of the technique.

Implicit method and slope limiter in AHMR procedure for high order discontinuous Galerkin methods for compressible flows

In this paper the robustness and the performance of adaptive hierarchical mesh refinement (AHMR) for high order Discontinuous Galerkin (DG) finite element method with slope limiting procedure combined with an implicit time scheme for the 2D non-linear Euler equations are shown. A slope limiting procedure based on triangular meshes is implemented and has been extended and amended accordingly to suit quadrilateral elements. The combination of DG methods and slope limiters is generally used with explicit time schemes. Here, the slope limiter implemented is incorporated into a quasi implicit time scheme procedure combined with an automatic h-adaptive hierarchical mesh refinement allowing non-conforming meshes. The time scheme is the implicit Second Order Backward Difference Formula (BDF2) with varying time step. The numerical test cases including subsonic, transsonic and supersonic flows show that the current slope limiting with quadrilateral meshes process together with the implicit time scheme is able to remove overshoots and undershoots around high gradient regions while preserving the high accuracy of the DG method. While combining this procedure with the automatic h-adaptive mesh refinement, one can improve the accuracy of the solutions and be able to capture quite precisely the features of the flows under consideration. The AHMR automatic procedure presented can easily be implemented in the numerical resolution of any physical models. Furthermore the limiting method used in this paper can be generalized to any type of mesh in two dimensions.

Isogeometric analysis of shear refined delaminated composite beams using dimensionally reduced beam sectional analysis

In this paper the isogeometric method is used to present a shear refined composite beam model through the concept of dimensional reduction method. A one-dimensional (1D) beam model is extracted from three-dimensional (3D) elasticity problem. The 1D beam model is developed using cross-sectional properties from two-dimensional (2D) beam sectional analysis. The 2D analysis of the cross-section is presented by implementing the transverse shear effects in isogeometric analysis. In beam cross-sectional problem, the influence of linear parameterization is investigated for isogeometric modeling of beam cross-section. Moreover, the effect of delamination on cross-sectional stiffness constants is discussed. Using isogeometric analysis (IGA), less degrees of freedom is needed in contrast to classical finite element method and automatic mesh refinement capability is attained. The present composite beam model eliminates the expensive use of 3D finite element analysis with its high precision and fidelity to 3D problem.