Adaptive Refinement Algorithm Research Articles

Improvement in the Accuracy and Efficiency of Smoothed Particle Hydrodynamics: Point Generation and Adaptive Particle Refinement/Coarsening Algorithms

Improvement in the Accuracy and Efficiency of Smoothed Particle Hydrodynamics: Point Generation and Adaptive Particle Refinement/Coarsening Algorithms

Globally optimal scheduling of an electrochemical process via data-driven dynamic modeling and wavelet-based adaptive grid refinement

Electrochemical recovery of succinic acid is an electricity intensive process with storable feeds and products, making its flexible operation promising for fluctuating electricity prices. We perform experiments of an electrolysis cell and use these to identify a data-driven model. We apply global dynamic optimization using discrete-time Hammerstein–Wiener models to solve the nonconvex offline scheduling problem to global optimality. We detect the method’s high computational cost and propose an adaptive grid refinement algorithm for global optimization (AGRAGO), which uses a wavelet transform of the control time series and a refinement criterion based on Lagrangian multipliers. AGRAGO is used for the automatic optimal allocation of the control variables in the grid to provide a globally optimal schedule within a given time frame. We demonstrate the applicability of AGRAGO while maintaining the high computational expenses of the solution method and detect superior results to uniform grid sampling indicating economic savings of 14.1%.

Agglomeration of polygonal grids using graph neural networks with applications to multigrid solvers

Agglomeration-based strategies are important both within adaptive refinement algorithms and to construct scalable multilevel algebraic solvers. In order to automatically perform agglomeration of polygonal grids, we propose the use of Machine Learning (ML) strategies, that can naturally exploit geometrical information about the mesh in order to preserve the grid quality, enhancing performance of numerical methods and reducing the overall computational cost. In particular, we employ the k-means clustering algorithm and Graph Neural Networks (GNNs) to partition the connectivity graph of a computational mesh. Moreover, GNNs have high online inference speed and the advantage to process naturally and simultaneously both the graph structure of mesh and the geometrical information, such as the areas of the elements or their barycentric coordinates. These techniques are compared with METIS, a standard algorithm for graph partitioning, which is meant to process only the graph information of the mesh. We demonstrate that performance in terms of quality metrics is enhanced for ML strategies. Such models also show a good degree of generalization when applied to more complex geometries, such as brain MRI scans, and the capability of preserving the quality of the grid. The effectiveness of these strategies is demonstrated also when applied to MultiGrid (MG) solvers in a Polygonal Discontinuous Galerkin (PolyDG) framework. In the considered experiments, GNNs show overall the best performance in terms of inference speed, accuracy and flexibility of the approach.

Implementation of the adaptive phase-field method with variable-node elements for cohesive fracture

In this paper, an adaptive phase-field model based on variable-node elements and error-indicator is presented to predict cohesive fracture evolution. The phase-field cohesive-zone model is characterized by the parameterized energetic degradation and crack geometric functions. A staggered iteration scheme is used to solve the coupled non-linear system of displacement field and phase field. The adaptive local refinement algorithm facilitated with error-indicator and variable-node elements is developed to improve computational efficiency. To determine the adaptive local refinement regions, we apply the phase-field threshold to characterize the crack regions and a novel history field threshold to denote crack tips. The variable-node elements are acted as transition elements to flexibly yet simply link the fine and coarse meshes. In addition, a fast method to generate the variable-node elements is given. Several representative numerical examples are studied to demonstrate the performance and accuracy of the proposed adaptive phase-field model, consisting of a mixed-model fracture of L-shape panel, the wedge-splitting test, the classic three point bending experiment and a plate with a hole and multi-cracks under tensile/shear loadings. The computer codes can be accessed at: https://github.com/hhuztc/Adaptive-PFM.git.

A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager–Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains.

Adaptive Nonintrusive Reconstruction of Solutions to High-Dimensional Parametric PDEs

Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a nonintrusive generalization of the adaptive Galerkin finite element method with residual-based error estimation. It combines the nonintrusive character of a randomized least squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the nonintrusive adaptive algorithm, showing the expected convergence rates of single-level strategies.

H-Adaptive radial basis function finite difference method for linear elasticity problems

In this research work, the radial basis function finite difference method (RBF-FD) is further developed to solve one- and two-dimensional boundary value problems in linear elasticity. The related differentiation weights are generated by using the extended version of the RBF utilizing a polynomial basis. The type of the RBF is restricted to polyharmonic splines (PHS), i.e., a combination of the odd m-order PHS phi (r)=r^m with additional polynomials up to degree p will serve as the basis. Furthermore, a new residual-based adaptive point-cloud refinement algorithm will be presented and its numerical performance will be demonstrated. The computational efficiency of the PHS RBF-FD approach is tested by means of the relative errors measured in ell _2-norm on several representative benchmark problems with smooth and non-smooth solutions, using h-adaptive, uniform, and quasi-uniform point-cloud refinement.

Accelerated Adaptive Error Control and Refinement for SIE Scattering Problems

We present an application of adjoint-based adaptive error control and refinement for scattering problems solved using the method of moments (MoM) in the electric field integral equation (EFIE) and the coupled EFIE and magnetic field integral equation (MFIE) formulations. Specifically, we examine the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) formulation of the EFIE-MFIE. We first construct the adjoint problems of the EFIE and the EFIE-MFIE for estimating the error of radar cross section (RCS) quantities of interest (QoIs) directly. We then introduce an effective adaptive refinement algorithm based on an error prediction heuristic and a posteriori error estimation, which enables rapid and consistent convergence regardless of a chosen tolerance and the coarseness of the starting discretization. The approach, moreover, inherently promotes equilibration of the QoI error contributions and produces, therefore, consistently balanced meshes. Numerical examples with canonical scattering targets and adaptive <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> -refinement confirm the strength of the proposed refinement method, demonstrating the ability to generate high-quality discretizations, both in terms of accuracy and efficiency, without expert user intervention.

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

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

Adaptive refinement for unstructured T-splines with linear complexity

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement scheme. We combine these ideas with an edge subdivision routine that allows for I-nodes, yielding a very flexible refinement scheme that nicely distributes the T-nodes, preserving global linear independence, analysis-suitability (local linear independence) except in the vicinity of extraordinary nodes, sparsity of the system matrix, and shape regularity of the mesh elements. Further, we show that the refinement procedure has linear complexity in the sense of guaranteed upper bounds on a) the distance between marked and additionally refined elements, and on b) the ratio of the numbers of generated and marked mesh elements.

Parallel adaptive weakly-compressible SPH for complex moving geometries

The use of adaptive spatial resolution to simulate flows of practical interest using Smoothed Particle Hydrodynamics (SPH) is of considerable importance. Recently, Muta and Ramachandran [1] have proposed an efficient adaptive SPH method which is capable of handling large changes in particle resolution. This allows the authors to simulate problems with much fewer particles than was possible earlier. The method was not demonstrated or tested with moving bodies or multiple bodies. In addition, the original method employed a large number of background particles to determine the spatial resolution of the fluid particles. In the present work we establish the formulation's effectiveness for simulating flow around stationary and moving geometries. We eliminate the need for the background particles in order to specify the geometry-based or solution-based adaptivity and we discuss the algorithms employed in detail. We consider a variety of benchmark problems, including the flow past two stationary cylinders, flow past different NACA airfoils at a range of Reynolds numbers, a moving square at various Reynolds numbers, and the flow past an oscillating cylinder. We also demonstrate different types of motions using single and multiple bodies. The source code is made available under an open source license, and our results are reproducible. Program summaryProgram Title: Parallel adaptive EDAC-SPHCPC Library link to program files:https://doi.org/10.17632/62rhbb9p4t.1Developer's repository link:https://gitlab.com/pypr/asph_motion.Code Ocean capsule:https://codeocean.com/capsule/4785489Licensing provisions: BSD 3-clauseProgramming language: PythonExternal routines/libraries:: PySPH (https://github.com/pypr/pysph), matplotlib (https://pypi.org/project/matplotlib/), NumPy (https://pypi.org/project/numpy/), automan (https://pypi.org/project/automan/), compyle (https://pypi.org/project/compyle/).Nature of problem: Simulating fluid flow around multiple moving bodies requires the local resolution to be automatically adapted in order to capture all the necessary flow features. If a single resolution is used throughout the domain, the number of particles would be excessively large.Solution method: We validate and demonstrate the accuracy of the adaptive particle refinement algorithm in simulating flow past moving multiple geometries. Our algorithm is fully parallel, and we provide an open-source implementation.

On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

Goal-oriented mesh adaptivity for inverse problems in linear micromorphic elasticity

In this work, we extend goal-oriented mesh adaptivity to parameter identification for a class of linear micromorphic elasticity problems. Starting from a compact formulation in our previous work (Ju and Mahnkhen, 2017), we propose a two-level optimization framework based on goal-oriented error estimation. By means of a sensitivity analysis of the generalized constitutive relations, we establish a gradient-based solver for the inverse problem, where parameters are optimized within an inner optimization loop for a given mesh. Exact error representations are derived from a Lagrange method, aiming at a quantity of interest as a user-defined functional of the parameters. By using a patch recovery technique for enhanced solutions, a computable error estimator is presented and used to drive an adaptive refinement algorithm, which forms an outer optimization loop. For a numerical study, we investigate the performance of the resulting adaptive procedure in case of perfect, incomplete and perturbed data. The results confirm the effectiveness of the proposed adaptive procedure.

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energy norms for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

A framework for efficient isogeometric computations of phase-field brittle fracture in multipatch shell structures

We present a computational framework for applying the phase-field approach to brittle fracture efficiently to complex shell structures. The momentum and phase-field equations are solved in a staggered scheme using isogeometric Kirchhoff–Love shell analysis for the structural part and isogeometric second- and fourth-order phase-field formulations for the brittle fracture part. For the application to complex multipatch structures, we propose penalty formulations for imposing all the required interface constraints, i.e., displacement (C0) and rotational (C1) continuity for the structure as well as C0 and C1 continuity for the phase field, where the latter is required only in the case of the fourth-order phase-field model. All involved penalty terms are scaled with the corresponding problem parameters to ensure a consistent scaling of the penalty contributions to the global system of equations. As a consequence, all coupling terms are controlled by one global penalty parameter, which can be set to 103 independent of the problem parameters. Furthermore, we present a multistep predictor–corrector algorithm for adaptive local refinement with LR NURBS, which can accurately predict and refine the region around the crack even in cases where fracture fully develops in a single load step, such that rather coarse initial meshes can be used, which is essential especially for the application to large structures. Finally, we investigate and compare the numerical efficiency of loosely vs. strongly staggered solution schemes and of the second- vs. fourth-order phase-field models.

Adjoint-Based Accelerated Adaptive Refinement in Frequency Domain 3-D Finite Element Method Scattering Problems

We present the application of adjoint analysis to 3-D finite-element method scattering problems for a posteriori error estimation and adaptive refinement. Adjoint-based methodologies, though underutilized in computational electromagnetics (CEM), enable significant improvements for both efficiency and accuracy. We first formulate the adjoint problem of the 3-D double-curl wave equation and the error estimates for the construction of novel accelerated adaptive refinement algorithms. We demonstrate adaptive error control for a customizable quantity of interest (QoI) resulting in targeted refinement and improved resource allocation through the application of automatic global and local error tolerance heuristics that accelerate the refinement process. The proposed refinement algorithms rapidly refine even extremely coarse initial discretizations to high accuracy, eliminating or substantially reducing manual intervention in the generation of computationally efficient and accurate simulations. Moreover, comparisons with analytical results validate our approach to accelerating automatic refinement to fine tolerances.

Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme

In this article adaptive refinement algorithms are presented to solve Poisson problems by a radial basis function partition of unity (RBF-PU) collocation scheme. Since in this context the problem of constructing an adaptive discretization method to be really effective is still open, we propose some error indicators and refinement strategies, so that each of these two essential ingredients takes advantage of the potentiality of the other one. More precisely, the refinement techniques coupled with a local error indicator is an ad-hoc strategy for the RBF-PU method. The resulting scheme turns out to be flexible and the use of efficient searching procedures enables us a fast detection of the regions that adaptively need the addition/removal of points. Several numerical experiments and applications support our study by illustrating the performance of our adaptive algorithms.

Asymmetric Adaptive Particle Refinement in SPH and Its Application in Soil Cutting Problems

In this study, a novel asymmetric adaptive particle refinement algorithm in smoothed particle hydrodynamics (SPH) is developed for soil cutting problems. Each candidate particle that located at the cutting blade of the structure is split into two “children” particles to minimize the oscillation of the contact force. And thus reduce the local instability. To minimize the density refinement error, a numerical method to determine the optimal smoothing lengths for “children” particles is given. To verify the accuracy of proposed algorithm, the adaptive refinement procedure are implemented into two models: one for soil cutting test on plane strain condition and the other for sample drilling test on axisymmetric condition. The observed flow pattern of the soil and contact forces are compared with laboratory experimental data available in the literature. Results indicate that the proposed asymmetric adaptive refinement algorithm could significantly avoid severe local instability and contributes to high-accuracy simulation.

Optimal Control Algorithms with Adaptive Time-Mesh Refinement for Kite Power Systems

This article addresses the problem of optimizing electrical power generation using kite power systems (KPSs). KPSs are airborne wind energy systems that aim to harvest the power of strong and steady high-altitude winds. With the aim of maximizing the total energy produced in a given time interval, we numerically solve an optimal control problem and thereby obtain trajectories and controls for kites. Efficiently solving these optimal control problems is crucial when the results are used in real-time control schemes, such as model predictive control. For this highly nonlinear problem, we derive continuous-time models—in 2D and 3D—and implement an adaptive time-mesh refinement algorithm. By solving the optimal control problem with such an adaptive refinement strategy, we generate a block-structured adapted mesh which gives results as accurate as those computed using fine mesh, yet with much less computing effort and high savings in memory and computing time.

Reliable anisotropic-adaptive discontinuous Galerkin method for simplified PN approximations of radiative transfer

In this paper we present the analysis for the error estimator for radiative transfer problems presented in Giani and Seaid (2016) where we showed the capabilities of the error estimator to accurately drive the adaptivity to resolve steep boundary layers, which are among the difficulties that most numerical methods fail to resolve accurately. We prove reliability for the error estimator in terms of a global upper bound of the error measured in the natural norm. We present a series of numerical experiments to test the efficiency of this approach within a fully automated h p -adaptive refinement algorithm.