Adaptive Refinement Strategy Research Articles

Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria

The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.

A comparative assessment of different adaptive spatial refinement strategies in phase-field fracture models for brittle fracture

For the smeared approximation of a discrete crack, phase-field fracture simulations of brittle materials require suitable finite element meshes in regions where crack propagation is expected to get an accurate resolution of the phase-field function. The straightforward option is to pre-refine the mesh in regions of the expected crack paths. However, this could lead to very computationally intensive simulations due to the high number of elements. Alternatively, adaptive spatial refinement of the finite element mesh is utilized based on appropriate error indicators to obtain the required accuracy in the areas of crack propagation. Different error indicators can be used: the most common one for phase-field fracture simulations is the threshold-based approach, in which elements are refined depending on the value of the phase-field function. Alternatively, the Kelly error indicator can be used as a criterion for spatial adaptivity. It considers the jumps in the gradients of the phase-field function between the elements. We additionally introduce here an error indicator based on configurational forces, that depend on the Eshelby stress tensor. For mode I loading in linear elastic fracture mechanics, the configurational forces have a close connection to the J-Integral and the critical fracture energy Gc, respectively. Therefore, a suitable norm of the configurational forces is introduced as an error indicator here. These three error indicators are introduced and compared to each other in terms of accuracy and efficiency by means of numerical examples for crack growth in the single edge notched shear test.

An Adaptive Refinement Scheme for Depth Estimation Networks

Deep learning has proved to be a breakthrough in depth generation. However, the generalization ability of deep networks is still limited, and they cannot maintain a satisfactory performance on some inputs. By addressing a similar problem in the segmentation field, a feature backpropagating refinement scheme (f-BRS) has been proposed to refine predictions in the inference time. f-BRS adapts an intermediate activation function to each input by using user clicks as sparse labels. Given the similarity between user clicks and sparse depth maps, this paper aims to extend the application of f-BRS to depth prediction. Our experiments show that f-BRS, fused with a depth estimation baseline, is trapped in local optima, and fails to improve the network predictions. To resolve that, we propose a double-stage adaptive refinement scheme (DARS). In the first stage, a Delaunay-based correction module significantly improves the depth generated by a baseline network. In the second stage, a particle swarm optimizer (PSO) delineates the estimation through fine-tuning f-BRS parameters-that is, scales and biases. DARS is evaluated on an outdoor benchmark, KITTI, and an indoor benchmark, NYUv2, while for both, the network is pre-trained on KITTI. The proposed scheme was effective on both datasets.

Active Learning Strategy for Surrogate-Based Quantile Estimation of Field Function

Uncertainty quantification is widely used in engineering domains to provide confidence measures on complex systems. It often requires to accurately estimate extreme statistics on computationally intensive black-box models. In case of spatially or temporally distributed model outputs, one valuable metric results in the estimation of extreme quantile of the output stochastic field. In this paper, a novel active learning surrogate-based method is proposed to determine the quantile of an unidimensional output stochastic process with a confidence measure. This allows to control the error on the estimation of a extreme quantile measure of a stochastic process. The proposed approach combines dimension reduction techniques, Gaussian process and an adaptive refinement strategy to enrich the surrogate model and control the accuracy of the quantile estimation. The proposed methodology is applied on an analytical test case and a realistic aerospace problem for which the estimation of a flight envelop is of prime importance for launch safety reasons in the space industry.

Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part I: Single-Level Approximation

A general adaptive refinement strategy for solving linear elliptic partial differential equations with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile [SIAM J. Numer. Anal., 56 (2018), pp. 3121--3143] to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available on GitHub.

Research on efficient and accurate computing strategies for complex optimal control problems

In this study, grid refinement strategies are studied to improve computing accuracy and efficiency of the complex optimal control problems. Firstly, a solution method based on multiple variable time nodes was proposed for the optimal control problem with Bang-Bang characteristics. The problem was transformed into the Mayer formula by adding relaxed variables to eliminate integral terms. And the implementation procedure for the algorithm was also defined. Then, for optimal control problems with continuous slope changes and key switching points, a slope-based adaptive grid refinement strategy was also proposed. The time grids were able to automatically increase or decrease based on the slope information, and a new technique was used to judge the switching points and refine the corresponding time grids. Finally, the performances of the proposed methods were verified with six classical examples. The results demonstrate that they comprehensively surpass traditional methods in computing accuracy and efficiency.

Adaptive control arc length-based time grid refinement control parameterisation method for unmanned hypersonic vehicle reentry trajectory optimisation

An adaptive control arc length-based trajectory optimisation approach, which combines the control vector parameterisation (CVP) method with Gauss time grid distribution and adaptive grid refinement strategy, is proposed to further improve the downrange of unmanned hypersonic vehicles. Firstly, the Gauss time grid strategy is introduced to obtain a nonuniform Gauss time grid to calculate the original control profiles. By employing the curve integral formula to analyse the control arc length, an adaptive time grid refinement concept is proposed and the refinement process is derived in detail. The initial time grid nodes are then adaptively subdivided and combined. Finally, the control arc length-based nonuniform time grid refinement optimisation method is implemented. Numerical tests are carried out on common aero hypersonic vehicles and comparisons are made with other CVP methods. The test results reveal that the proposed approach can efficiently improve the downrange and decrease the path constraint oscillation.

Flux recovery scheme for elliptic interface problems

This paper introduces a flux recovery algorithm that ensures the continuity of the flux at the interface and improves its accuracy. At each subdomain created by an interface, the gradient is recovered using the polynomial preserving recovery method and a weight is given to each subdomain based on these recovered gradients. Then from these recovered gradients and subdomain weights, we form what we call a corrected recovered gradient that is exploited to construct the recovered flux at the interface. An error estimator that is based on the recovered flux is formulated for an adaptive refinement scheme that improves the accuracy of the flux near the interface. We present the numerical results of applying the proposed scheme to some two-dimensional experiments with different types of interfaces.

A phase field method with adaptive refinement strategy and virtual crack insertion technique

In this paper a novel adaptive refinement strategy, which features no hanging nodes or hybrid elements generated, is proposed for phase-field model to realize the refinement of 2-D quadrilateral elements. The strategy is implemented and proved the capacity of enhancing efficiency in calculation. Meanwhile, the virtual crack insertion technique is developed to simplify mesh generation process in 2-D and 3-D phase-field based fracture models. The technique is also proved to be robust in simulating the initial crack with trial on benchmark examples. The applicability of phase-field method with novel adaptive refinement strategy and virtual crack insertion technique for fracture is demonstrated by numerical experiments of 2-D models containing single and multiple cracks. The work shows great promise in engineering application. In addition, mode I-II-III fracture can be well simulated with virtual crack insertion technique. Therefore, virtual crack is recommended on fracture simulation of 3-D problems for convenient mesh generation and higher computational efficiency.

Moving level-of-detail surfaces

We present a simple, fast, and smooth scheme to approximate Algebraic Point Set Surfaces using non-compact kernels, which is particularly suited for filtering and reconstructing point sets presenting large missing parts. Our key idea is to consider a moving level-of-detail of the input point set which is adaptive w.r.t. to the evaluation location, just such as the samples weights are output sensitive in the traditional moving least squares scheme. We also introduce an adaptive progressive octree refinement scheme, driven by the resulting implicit surface, to properly capture the modeled geometry even far away from the input samples. Similarly to typical compactly-supported approximations, our operator runs in logarithmic time while defining high quality surfaces even on challenging inputs for which only global optimizations achieve reasonable results. We demonstrate our technique on a variety of point sets featuring geometric noise as well as large holes.

An adaptive floating node based formulation for the analysis of multiple delaminations under high cycle fatigue loading

A novel efficient numerical formulation for the analysis of multiple fatigue-driven delamination cracks is presented. A cohesive zone model is used in combination with an Adaptive Refinement Scheme (ARS) and an Adaptive Floating Node Method (A-FNM) element that refine the model effectively during the analysis. Novel techniques are proposed to track the positions of multiple crack tips and calculate the mode decomposed energy release rates for the individual crack tips using the J-integral. The method has been implemented in a Matlab finite element code and validated with single and multiple delamination cases with varying mode mixities. Comparisons with theoretically based predictions and available experimental data showcase the high accuracy of the method. The presented method lowers the computational time compared to standard, fully refined finite element models by a factor of 4–5.

Geometrically nonlinear design of compliant mechanisms: Topology and shape optimization with stress and curvature constraints

In this work, we exploit a sequential topology and shape optimization framework to design compliant mechanisms. We pay particular attention to a refined and exact design of flexure hinges via shape optimization in a geometrically nonlinear setting. We propose improved adaptive shape and domain refinement strategies for the embedding domain discretization method (EDD) to achieve numerically and geometrically accurate designs of the flexure hinges. Furthermore, to account for durability and manufacturability of the flexure hinges, we employed local stress constraints and a curvature constraint. The local stress constraints are integrated into an augmented Lagrange functional together with the output displacement to form a combined objective response. The design update is realized by employing and adapting the traction method for the specifics of the embedded boundary. Hence, the curvature constraint does not appear in a standard response form, but is rather introduced into the auxiliary boundary value problem (BVP) of the traction method via a penalty functional. The novelties of this work include: (1) enhanced adaptive shape and domain refinement strategies for EDD; (2) geometrically nonlinear shape optimization using EDD; (3) adapted traction method with curvature constraint.

Sequential topology and shape optimization framework to design compliant mechanisms with boundary stress constraints

We present a sequential topology and shape optimization framework to design compliant mechanisms with boundary stress constraints. In our approach, a density-based topology optimization method is used to generate the configuration of the mechanisms. Afterwards, a node-based shape optimization is invoked to obtain an exact boundary representation. A specialized, optimality criteria-based design update is formulated for the shape optimization. To avoid impractical hinges with point connections, stress constraints are imposed. The stress constraints are imposed using two strategies: Local stress constraints on the nodes of the boundary or global P-norm stress constraints in the domain. Further, an adaptive shape refinement strategy is adopted to increase the design space of shape optimization and to capture the fine-scale details of the geometry. Finally, numerical experiments are presented, showing that the proposed approach can be effectively applied to the design of compliant mechanisms with stress constraints.

Investigation on the bouncing and coalescence behaviors of bubble pairs based on an improved APR-SPH method

Bubbles rising in fluids are widespread in many engineering fields. In this paper, the bouncing and coalescing of two bubbles rising side by side are studied based on a multi-phase smoothed particle hydrodynamics (SPH) model. An improved adaptive particle refinement (APR) scheme is applied in the SPH model to make the numerical simulation sufficiently accurate and efficient. The effect of the surface-tension model is firstly verified, then various testing cases including the rising of a single bubble and two horizontally positioned bubbles are carried out and validate the numerical model well. Furthermore, the interaction between two three-dimensional bubbles rising side by side is studied considering the effects of viscous forces (with the Reynolds number Re), surface tensions (with the Eotvos number Eo), and the initial centroid distance of the two bubbles (Γ). The motion and the deformation of the bubbles are displayed and the mechanism of the bubbles' interaction is discussed. The results reveal that for the lower Eotvos number cases indicating a larger surface tension effect, the surface tension force increases dramatically when the two bubbles collide with each other and coalesce.

An adaptive floating node based formulation for the analysis of multiple delaminations under quasi-static loading

A novel and efficient numerical formulation for the modelling of multiple delaminations growth in laminated composite materials subjected to quasi-static loading is presented. The proposed formulation alleviates the high computational cost associated with models featuring cohesive elements by using a novel Adaptive Refinement Scheme and an Adaptive Floating Node Method Element to refine the model effectively during the analysis without modifying the global finite element connectivity. The formulation has been implemented in a MATLAB finite element code and validated with single and multiple delamination numerical models with varying mode mixities. The new formulation provides accurate results comparable to standard fully refined finite element models while drastically lowering the computational time of the analysis.

On Solving Semilinear Singularly Perturbed Neumann Problems for Multiple Solutions

Based on the analysis of bifurcation points and Morse indices of trivial solutions at any perturbation value for a general semilinear singularly perturbed Neumann boundary value problem, in this paper, the exact critical perturbation value $\varepsilon_c$ which determines the existence or nonexistence of nontrivial positive solutions is obtained. As a result, the generating process of nontrivial positive solutions is studied and further used to guide algorithm design and numerical computation. An improved local minimax method is then proposed accordingly to compute both M-type and W-type saddle points by using an adaptive local refinement strategy and a Newton method to overcome singularity difficulty and accelerate local convergence. Extensive numerical results are reported to justify the critical perturbation value $\varepsilon_c$ and investigate some interesting solution properties of different types of problems.

Adaptive finite element modeling of phase-field fracture driven by hydrogen embrittlement

The effect of hydrogen on crack propagation has been studied extensively using various numerical algorithms. Recently, phase-field models have been developed that can predict crack propagation with very good resemblance of the crack paths with the experimental results. Solving the coupled equations of hydrogen transport, mechanical equilibrium and phase-field models is computationally expensive requiring a large number of time steps and high spatial resolution. Often, the spatial resolution is achieved a priori in known regions where the cracks may propagate. This leads to a large number of degrees of freedom to be solved for a large number of time steps for the coupled equations in simulating hydrogen-assisted cracking. In the present work, an adaptive refinement scheme is proposed that will remove the burden of very high uniform spatial resolution in non crack regions eventually leading to fewer degrees of freedom and hence decreases the computational cost. Several case studies are considered and the efficacy of the proposed method is demonstrated.

Spatial adaptivity in SOLEDGE3X‐HDG for edge plasma simulations in versatile magnetic and reactor geometries

AbstractWith the ultimate goal to predict plasmas heat and particle fluxes in ITER operation, more efforts are required to deal with realistic magnetic configurations and tokamak geometries. In an attempt to achieve this goal, we propose an adaptive mesh refinement method added to a fluid solver based on a high‐order hybrid discontinuous Galerkin (HDG) method. Based on unstructured meshes, this magnetic equilibrium free numerical scheme has shown promising and encouraging features to solve 2D/3D transport reduced Braginski fluid equations. To improve its numerical efficiency, a mesh refinement based on h‐adpativity is investigated. We describe here an adaptive refinement strategy on a reduced edge particle transport model based on electron density and parallel momentum. This strategy is illustrated in realistic tokamak wall geometry. Computations performed show potential gains in the required number of degrees of freedom against benchmark computations with uniform meshes, along with the potential to give an automated, goal‐oriented, mesh generation technique for edge transport simulations in 2D.

A Computational Model on Cartesian Adaptive Grid for Thrombosis Simulation

Computer simulation has been an effective experimental method to study the mechanism of thrombosis. As a computational model is the core of the simulation, a hybrid computational model which focuses on improving the modelling of thrombus growth is proposed in this paper. In the model, the process of thrombus growth is implemented by using Level Set Method that remains the particle morphology while converting particles into continuum materials. Furthermore, the model is constructed on a novel Adaptive Mesh Refinement strategy that enhances the efficiency of calculation and data storage. The proposed model is used in the simulation experiments of platelet thrombus formation in stenotic arteries to gained the influences of stenosis and blood flow velocity for thrombosis which are consistent with in vivo experiments. The simulations show that compared to refining the grid containing thrombus uniformly, the local level set based adaptive refinement strategy performs wall in reducing cell data.

Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the “shape” of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements' shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is independent of the differential model the overall computational costs are kept low.