Mesh Adaptation Strategy Research Articles

Theoretical and Numerical Analysis of Singular Perturbation Problems in Ordinary Differential Equations

Background: Singular perturbation troubles in everyday differential equations (ODEs) involve a small parameter ϵ that causes answers to show off behaviors on multiple scales, main to massive challenges in both theoretical and numerical evaluation. Objective: This paper at targets to comprehensively look at the theoretical and numerical techniques for reading and solving singular perturbation troubles in ODEs. Focus is located on know-how the behaviors precipitated through the small parameter and developing strong numerical techniques to accurately seize these behaviors. Methods: Theoretical approaches employed include matched asymptotic expansions and multiple scale analysis. These methods decompose the solution domain into regions with different scales, constructing and matching approximate solutions to ensure smooth transitions. Numerical techniques such as finite difference methods, finite element methods, and spectral methods are utilized, with particular emphasis on adaptive mesh refinement to handle boundary layers effectively Results: Theoretical evaluation demonstrates the effectiveness of matched asymptotic expansions and multiple scale analysis in presenting correct approximations and easy transitions among one of a kind solution regions. Numerical techniques showed high accuracy and performance, particularly whilst blended with adaptive techniques. Finite distinction strategies with non-uniform grids and adaptive mesh refinement, finite detail techniques with adaptive mesh strategies, and spectral strategies with special handling of boundary layers all proved successful. Stability and convergence analyses showed the reliability of those techniques. Conclusions: This comprehensive evaluation highlights the strengths and applicability of each theoretical and numerical strategies in tackling singular perturbation issues in ODEs. The mixed use of these techniques lets in for correct and green answers, presenting treasured equipment for applications in diverse clinical and engineering fields.

Adaptive Mesh Strategy for Efficient Use of Interface Elements in a 3D Probabilistic Explicit Cracking Model for Concrete.

In this paper, the development of a 3D adaptive probabilistic explicit cracking model for concrete is reported. The contribution offered herein consists in a new adaptive mesh strategy designed to optimize the use of interface elements in probabilistic explicit cracking models. The proposed adaptive mesh procedure is markedly different from other strategies found in the literature, since it takes into account possible influences on the redistribution of stresses after cracking and can also be applied to purely deterministic cracking models. The process of obtaining the most appropriate adaptive mesh procedure involved the development and evaluation of three different adaptivity strategies. Two of these adaptivity strategies were shown to be inappropriate due to issues related to stress redistribution after cracking. The validation results demonstrate that the developed adaptive probabilistic model is capable of predicting the scale effect at a level similar to that experimentally observed, considering the tensile failure of plain concrete specimens. The results also show that different softening levels can be obtained. The proposed adaptive mesh strategy proved to be advantageous, being able to promote significant reductions in the simulation time in comparison with the classical strategy commonly used in probabilistic explicit cracking models.

Non-uniform knot (NUK) SIAC post-processing of flow fields produced through unstructured grid adaptation and optimization

As the finite element method (FEM) and the finite volume method (FVM), both their traditional and high-order variants, continue their proliferation into various applied engineering disciplines, adaptive mesh refinement and optimization strategies have increased in their importance when solving real-world computational fluid mechanics applications. The post-processing and visualization of the resulting flow fields present two significant analysis and visualization challenges. The first challenge is the handling of elemental continuity, which is often only C0 continuous (in continuous Galerkin methods) or piecewise discontinuous (in discontinuous Galerkin methods). The second challenge is that, depending on the flow regime and the geometric configurations for which adaptive meshing strategies are used, the meshes generated are often highly anisotropic. The (uniform knot) line-SIAC (L-SIAC) filter has been proposed as a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous. In this paper, we demonstrate that the state-of-the-art adaptive L-SIAC filter, designed for mildly anisotropic meshes, suffers degradation in the quality of the post-processed solution when applied to the types of highly anisotropic meshes produced through adaptive mesh refinement and optimization. Hence, a new Non-Uniform Knot (NUK) L-SIAC filter is proposed that automatically conforms to the underlying mesh anisotropy. We demonstrate that the new filter behaves similarly to the adaptive L-SIAC filter when applied to uniform and mildly anisotropic meshes, and furthermore we show the superiority of the NUK L-SIAC filter when applied to highly anisotropic meshes. The newly formulated filter is applied to 2D canonical scalar fields and used to visualize 2D and 3D fluid flow simulation results.

A computational study for simulating MHD duct flows at high Hartmann numbers using a stabilized finite element formulation with shock-capturing

The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection–diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov–Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.

Modeling and simulation of electrochemical and surface diffusion effects in filamentary cation-based resistive memory devices

Cation-based (or electrochemical) resistive memory devices are gaining increasing interest in neuromorphic applications due to their capability to emulate the dynamic behavior of biological neurons and synapses. The utilization of such devices in neuromorphic systems necessitates a reliable physical model for the resistance switching mechanism, which is based on the formation and dissolution of a conductive filament in a thin dielectric layer, sandwiched between two metal electrodes. We propose a comprehensive model to simulate the evolution of the filament geometry under the effect of both surface diffusion caused by curvature gradient and electromechanical stress, and mass injection due to electrodeposition of cations. The model has been implemented in a C++ platform using a level-set approach based on a mixed finite element formulation, enriched by a mesh adaptation strategy to accurately and efficiently track the evolution of the filament shape. The numerical scheme is initially validated on various benchmark case studies. We then simulate the growth and self-dissolution of the filamentary geometry, incorporating an electrical model allowing a comparison with conventional cation-based memories. The simulations showcase filament formation under varying applied voltages and filament dissolution under different initial resistances.

Enhancing CFD-based design of wind turbine blades

CFD-based shape design is a state-of-the-art approach in wind energy that can lead to novel shape features. High computation costs have, however, until now hindered a widespread use of the approach. To address this issue, this work investigates two ways to enhance a CFD-based design framework, using a shape study of wind turbine blade tips as a test case. First, a quantification of the speed-up factor (× 3.6) achieved through so-called tailored meshes is presented. Then, a comparison of mesh adaptation strategies is given, further speeding up the procedure (× 2.3). The combined speed-up factor reaches above 8, showing that CFD-based shape design procedures may indeed be accelerated using tailored meshes and a correct mesh adaptation strategy.

An adaptive mesh scheme of the lattice spring model based on geometrical continuity

An adaptive mesh scheme is introduced for the lattice spring model (LSM), where the original triangular cells are subdivided into a set of smaller triangular cells. The scheme is based on geometrical continuity at the heterogeneous mesh boundary, where the refined grid cells intersect the original cell edge. The LSM simulations on the refined grid show a superior computational efficiency to the uniform grid. Each subdivision reduces the original cell edges by a factor of two. The refinement procedure was recursively applied ten times before any marked loss in accuracy was observed. The accuracy of the adaptive model is on par with a regular grid approach. More specifically, the characteristics of fracture cavity are comparable with a uniform grid of the same mesh density as the smallest cells in the adaptive approach. The fracture criterion such as J-integral, the elastic energy of the grid and potential energy change due to fracture growth and strain loading agree well with the theory of a mode I fracture, which enables simulations of process such as sub-critical fracture with a wide dynamic range.

SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing

This study is interested in stabilized finite element computations of advection-dominated convection–diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov–) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov–Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZβ technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZβ combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.

Adaptive scaled boundary finite element method for two/three-dimensional structural topology optimization based on dynamic responses

This paper presents an efficient solution for designing the topology of structures that can withstand dynamic loads. The method, called image/stereolithography (STL)-based adaptive scaled boundary finite element (SBFE), is a novel approach to topology optimization (TO) that is particularly effective for designing two/three-dimensional structures. The SBFE-based TO algorithm adopts a bi-evolutionary structural optimization (BESO) approach, which utilizes image/STL-based procedures to automatically adapt meshes and minimize mechanical compliance under varying volume fractions in the time domain. A key advantage of this approach is that it uses images of TO responses, rather than specific error functions, to guide the automatic adaptive mesh schemes. The method also incorporates a quadtree/octree mesh refinement to address the problem of hanging nodes, ensuring fast mesh convergence during the optimization process. Furthermore, a high-order implicit time integration technique provides a close approximation of equivalent static loads, allowing for the accurate mapping of displacement fields under actual dynamic responses at each time step. The effectiveness of this method is demonstrated through a variety of numerical examples and benchmarks.

A posteriori error analysis and mesh adaptivity for a virtual element method solving the Stokes equations

We investigate an adaptive mesh strategy for the conforming virtual element method (VEM) of the Stokes equations proposed in Manzini and Mazzia (2022). The VEM generalizes the finite element approach to polygonal and polyehedral meshes in the framework of Galerkin approximation. The scheme of Manzini and Mazzia (2022) is inf-sup stable, converges optimally in the L2 and energy norm for all polynomial orders k≥1, and the Stokes velocity is weakly divergence-free at the machine precision level. Our adaptive mesh strategy is based on a suitable residual-based a posteriori indicator. A posteriori analysis shows that such an indicator is theoretically efficient and reliable. Our numerical experiments show that it can be an efficient tool for solving scientific and engineering problems by applying it to a set of representative situations, including the case of a weakly singular solution as that of the “L-shape” domain.

Automatic variationally stable analysis for finite element computations: Transient convection-diffusion problems

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time “slices” in an attempt to reduce the computational cost of the space-time discretizations.We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies.

Tidal turbine array modelling using goal-oriented mesh adaptation

To examine the accuracy and sensitivity of tidal array performance assessment by numerical techniques applying goal-oriented mesh adaptation. The goal-oriented framework is designed to give rise to adaptive meshes upon which a given diagnostic quantity of interest (QoI) can be accurately captured, whilst maintaining a low overall computational cost. We seek to improve the accuracy of the discontinuous Galerkin method applied to a depth-averaged shallow water model of a tidal energy farm, where turbines are represented using a drag parametrisation and the energy output is specified as the QoI. Two goal-oriented adaptation strategies are considered, which give rise to meshes with isotropic and anisotropic elements. We present both fixed mesh and goal-oriented adaptive mesh simulations for an established test case involving an idealised tidal turbine array positioned in a channel. With both the fixed meshes and the goal-oriented methodologies, we reproduce results from the literature which demonstrate how a staggered array configuration extracts more energy than an aligned array. We also make detailed qualitative and quantitative comparisons between the fixed mesh and adaptive outputs. The proposed goal-oriented mesh adaptation strategies are validated for the purposes of tidal energy resource assessment. Using only a tenth of the number of degrees of freedom as a high-resolution fixed mesh benchmark and lower overall runtime, they are shown to enable energy output differences smaller than 2% for a tidal array test case with aligned rows of turbines and less than 10% for a staggered array configuration.

Adaptive Mesh Refinement Strategies for Cost-Effective Eddy-Resolving Transient Simulations of Spray Dryers

The use of adaptive meshing strategies to perform cost-effective transient simulations of spray drying processes is evaluated. These simulations are often computationally expensive, given the large differences between the characteristic times of the central jet and those of the unsteady flow developed at its periphery. Managing the computational cost through the control of the grid resolution by regions is inadequate in many of these applications since the grid resolution requirements change dynamically within the domain. These conditions are related to the unsteady nature of the flow in both the central jet and the flow recirculation zones. Therefore, the application of adaptive mesh refinement (AMR) strategies is recommended. In this paper, general AMR criteria based on relative errors are evaluated by testing three mesh adaptation criteria: velocity gradient, pressure gradient, and vorticity. This evaluation is performed using a low-cost turbulence model with eddy resolution (DDES) in two different types of drying chambers, in which experimental measurements are available. The use of AMR exerts appreciable effects on decreasing computational costs and contributes to the capture of large eddies in critical regions. The present approach provides an appropriate balance between solution accuracy and computational cost. By using a correct AMR configuration, it is possible to obtain results similar to those obtained on a fixed grid but reducing the computational costs by 3 to 5 times.

Development of a fully automatic damage simulation framework for quasi-brittle materials

This paper proposes a fully automatic analysis framework for static damage analysis of quasi-brittle materials exploiting adaptive mesh refinement approaches as well as adaptive load stepping techniques. The damage process is modelled using an integral-type nonlocal damage model which alleviates the mesh dependence issue. The scaled boundary finite element method and quadtree meshing algorithm are employed in the adaptive analysis. The quadtree algorithm is simple for adaptively remeshing and the mesh refinement is limited to local regions around the damage zones. The elements of the quadtree mesh are modelled by the scaled boundary finite element method, naturally resolving the hanging nodes encountered by the standard finite elements. Furthermore, the regular element shapes and small number of unique element patterns permit a significant reduction of computational cost on processing the element formulations, and an easy implementation of data transfer between two successive meshes. In the last step, the adaptive meshing strategy is combined with an automatic load stepping scheme to create a fully automatic analysis framework. By means of various numerical examples the performance of the proposed method predicting the damage evolution in structures is analysed in terms of its accuracy and efficiency.

A Posterior h/p Adaptive Refinement Algorithm for 3-D Goal-Oriented Electromagnetic Problems

We propose a posterior h/p adaptive mesh refinement algorithm for 3-D goal-oriented electromagnetic wave modeling in complex media. This new algorithm allows both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$h$</tex-math> </inline-formula> -version refinement—mesh size adjustment—and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$</tex-math> </inline-formula> -version refinement—polynomial order adjustment—simultaneously. To be specific, we use the adjoint system on the dual space of the forward solver; as a result, error generation, propagation, and accumulation can be evaluated, even without the analytical solutions. In particular, the posterior error is explicitly indicated by postprocessing of original forward solver. Compared with conventional mesh adaptation strategies, such an algorithm is goal-oriented and auxiliary-system-free; as a result, it does not require pay extensive attention to entire computational domain and is complimentary in solving adjoint state equation. Numerical examples demonstrate the superior performance of the proposed mesh refinement algorithms for real-world applications, compared to uniform refinement, nongoal-oriented adaptation, and other goal-oriented adaptation.

An adaptive viscosity regularization approach for the numerical solution of conservation laws: Application to finite element methods

We introduce an adaptive viscosity regularization approach for the numerical solution of systems of nonlinear conservation laws with shock waves. The approach seeks to solve a sequence of regularized problems consisting of the system of conservation laws and an additional Helmholtz equation for the artificial viscosity. We propose a homotopy continuation of the regularization parameters to minimize the amount of artificial viscosity subject to positivity-preserving and smoothness constraints on the numerical solution. The regularization methodology is combined with a mesh adaptation strategy that identifies the shock location and generates shock-aligned meshes, which allows to further reduce the amount of artificial dissipation and capture shocks with increased accuracy. We use the hybridizable discontinuous Galerkin method to numerically solve the regularized system of conservation laws and the continuous Galerkin method to solve the Helmholtz equation for the artificial viscosity. We show that the approach can produce approximate solutions that converge to the exact solution of the Burgers' equation. Finally, we demonstrate the performance of the method on inviscid transonic, supersonic, hypersonic flows in two dimensions. The approach is found to be accurate, robust and efficient, and yields very sharp yet smooth solutions in a few homotopy iterations.

Design of innovative self-expandable femoral stents using inverse homogenization topology optimization

In this study, we propose a novel computational framework for designing innovative self-expandable femoral stents. First, a two-dimensional stent unit cell is designed by inverse homogenization topology optimization. In particular, the unit cell is optimized in terms of contact area with the target of matching prescribed mechanical properties. The topology optimization is enriched by an anisotropic mesh adaptation strategy, enabling a time- and cost-effective procedure that promotes original layouts. Successively, the optimized stent unit cell is periodically repeated on a hollow cylindrical surface to construct the corresponding three-dimensional device. Finally, structural mechanics and computational fluid dynamics simulations are carried out to verify the performance of the newly-designed stent. The proposed workflow is being tested through the design of five proof-of-concept stents. These devices are compared through specific performance evaluations, which include the assessments of the minimum requirement for usability, namely the ability to be crimped into a catheter, and the quantification of the radial force, the foreshortening, the structural integrity and the induced blood flow disturbances.

Seismic Bearing Capacity of Strip Footings Placed Adjacent to Rock Slopes

ABSTRACT The bearing capacity of strip footings placed adjacent to the rock slopes subjected to pseudo-static horizontal earthquake body forces has been investigated by using the upper bound (UB) finite element limit analysis (FELA) in combination with a mesh adaptive strategy. The generalized Hoek–Brown (GHB) failure criterion is applied to describe the strength properties of rock masses, which can be treated as its native form without the smoothing of the yield surface while employing the second-order cone and power cone programming to solve the optimization models. Based on the UB FELA method, the seismic bearing capacity factor is determined considering the effect of different governing parameters, namely the horizontal earthquake acceleration coefficient k h , the normalized slope height H/B, the rock strength ratio σ ci /γB, the geological strength index GSI and the material constant m i . For design purpose, a series of non-dimensional charts are provided, and typical failure mechanisms are also discussed in detail.

A dynamic multiscale model of cerebral blood flow and autoregulation in the microvasculature

Models of the micro-circulatory blood flow in the brain can play a key role in understanding the variety of cerebrovascular diseases that occur in the microvasculature. These conditions are often linked to structural modifications in the vessel network, alterations in the blood flow patterns, as well as impairment in the autoregulatory response, all of which are pathological changes that the model should be able to address if it were to have any clinical value. Furthermore, if the model results were to be validated against clinical MRI data, the model simulations need to be computationally feasible when used on networks on the scale of an MRI voxel. This requires some form of an upscaling approach that bypasses the need for an explicit architectural representation of the whole network while maintaining the relevant anatomical connections. To this end, we developed a hybrid multiscale model of blood flow and autoregulation that traces the dynamic changes in blood flow, volume, and pressure in the cortical microvasculature, where the discrete topology of the penetrating vessels is preserved, and these are then appropriately coupled to the homogenised capillary bed by a spatially distributing support function in the terminal endings. In contrast to the other multiscale models, the model developed here accounts for the dynamic physiological phenomena of the blood flow and the autoregulation processes in the microvessels. We show how the adaptive meshing scheme for the capillary bed developed in this study can be employed to ensure a scale-invariant coupling formulation and numerically accurate simulations, all without compromising the computational feasibility of the model. A statistically accurate cortical network on the scale of an MRI voxel is generated, and the model parameter values are calibrated using a Monte Carlo Filtering analysis to ensure that the model results are physiologically informed. The model is found to be able to capture the steep pressure gradients that have been reported to occur at coupling interfaces. Furthermore, in response to an upstream pressure drop, the network is found to be able to recover cerebral blood flow while exhibiting the characteristic autoregulatory behaviour in terms of changes in vessel calibre and the biphasic flow response. Overall, the model developed here offers a high-quality characterisation of dynamic flow and autoregulation in the microvasculature at improved computational efficiency and lays the ground for whole-brain dynamic simulations.

Unlined Length Effect on the Tunnel Face Stability and Collapse Mechanisms in c-ϕ Soils: A Numerical Study with Advanced Mesh Adaptive Strategies

This paper presents a stability study on the collapse mechanisms of a plane-strain tunnel face in c-ϕ soils using the upper bound finite element method with rigid translatory moving elements (UBFELA-RTME) and nonlinear programming technique. Practical considerations are given to the unlined length influence behind the tunnel face. An advanced mesh adaptive updating strategy is adopted, aiming to improve the computational efficiency, the accuracy of upper-bound solutions, as well as the produced collapse mechanisms. The unlined length influence on the face stability and collapse mechanism of the tunnel face are determined with various combinations of tunnel depth ratios, soil friction angles, and dilatancy angles. Using the UBFELA-RTME with the Davis’s approach and a mesh adapting strategy, the non-associated plasticity flow rule can be well approximated. The developed technique was validated against different numerical methods, and it is concluded that the tunnel face stability can be improved by increasing soil friction and dilatancy angles, and yet weakens as the unlined length increases where a mesh-liked collapse zone gradually appears on the tunnel vault top. It gradually evolves to a global collapse failure till the ground surface. The findings contribute to a better understanding of the ground surface failure under the unlined support length influence in tunnel construction.