Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines

The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.

In this paper, the refinement strategy based on the "Non-Linear Localized Full MultiGrid" solver originally published in Int. J. Numer. Meth. Engng 84(8):947---971 (2010) for 2-D structural problems is extended to 3-D simulations. In this context, some extra information concerning the refinement strategy and the behavior of the error indicators are given. The adaptive strategy is dedicated to the accurate modeling of elastoplastic materials with isotropic hardening in transient dynamics. A multigrid solver with local mesh refinement is used to reduce the amount of computational work needed to achieve an accurate calculation at each time step. The locally refined grids are automatically constructed, depending on the user prescribed accuracy. The discretization error is estimated by a dedicated error indicator within the multigrid method. In contrast to other adaptive procedures, where grids are erased when new ones are generated, the previous solutions are used recursively to reduce the computing time on the new mesh. Moreover, the adaptive strategy needs no costly coarsening method as the mesh is reassessed at each time step. The multigrid strategy improves the convergence rate of the non-linear solver while ensuring the information transfer between the different meshes. It accounts for the influence of localized non-linearities on the whole structure. All the steps needed to achieve the adaptive strategy are automatically performed within the solver such that the calculation does not depend on user experience. This paper presents three-dimensional results using the adaptive multigrid strategy on elastoplastic structures in transient dynamics and in a linear geometrical framework. Isoparametric cubic elements with energy and plastic work error indicators are used during the calculation.

Mesh refinement for anisotropic diffusion in magnetized plasmas

This study focuses on the entry guidance problem, which is a highly nonlinear and constrained optimal control problem. The entry guidance problem is formulated as a nonlinear trajectory optimization problem with six-degrees-of-freedom dynamics, path constraints, and boundary constraints. The formulated problem is discretized and solved using sequential convex programming (SCP) with successive linearization. However, during the discretization process, conventional collocation methods encounter a challenge with high-frequency jittering in the control and bank angle profiles near active path constraints. Because the considered problem especially involves tight path constraints, the solution jittering issue becomes severe. Therefore, this study proposes a novel mesh refinement strategy to address this problem. Unlike conventional mesh refinement strategies based on interpolation error, the proposed approach resolves the convergence and solution jittering issue of the SCP framework by linking the direct and indirect methods with the costate estimation under successive linearization. Along with the development, we introduce a novel costate estimation method and provide proof of its validity. We present numerical simulation results of the proposed method and conduct a comparison with conventional approaches. The results obtained demonstrate successful mitigation of the solution jittering issue and a significant improvement in solution accuracy regarding compliance with the necessary conditions of optimality.

A directional error estimator for adaptive limit analysis

We present a 3D forward modeling algorithm of marine controlled-source electromagnetic (CSEM) fields using finite element method based on unstructured meshes. As the unstructured mesh is incorporated, the algorithm can easily accommodate complex structures, such as dipping interfaces and rough topography. Secondary coupled-potential formulation of Maxwell's equations allows to avoid the singularities introduced by the sources. The primary electromagnetic fields excited by electric dipole source in layered isotropic media are derived from the semi-analytical formulas based on Schelkunoff potentials. To improve the accuracy and efficiency of the algorithm, we propose an effective local mesh refinement strategy, where the elements near the receiver locations and in the abnormal areas are refined using the volume constraints. We validate the finite element code against a 1D and 2D reservoir model and study the performance of the local mesh refinement strategy. The Canonical disc model illustrates the utility of the finite element method for 3D CSEM modeling. The bathymetry model study shows that the seafloor topography gives an important response which needs to be reproduced by numerical modeling to avoid the misinterpretation of measurements.

An efficient variable-node XFEM for modeling multiple crack growth: A Matlab object-oriented implementation

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

An explicit time integration algorithm to solve the two-dimensional Navier-Stokes equations on general N-sided polygons is presented. The integral form of the equations are descritized on a set of non-overlapping polygons yielding a data structure where the flow variables are stored at cell centers. An edge-base data structure is utilized to obtain maximum flexibility and allow the use of polygons with any number of faces. The edgebased data structure allowed the scheme to be extended to allow a local mesh refinement strategy in areas where flow solution errors were deemed to be the greatest. A Roe-Averaged Riemann solver is utilized to evaluate the numerical flux at the cell faces and higher order accuracy is achieved by linear reconstruction on the cell-averaged data and using a monotone preserving limiter to achieve smooth results near flow discontinuities.

This paper aims to apply the Isogeometric Analysis(IGA) to fluid-structure interaction problem in the civil engineering field. Recently, IGA has attracted much attention as an analysis method to structure with arbitrary surfaces. In this paper, IGA is applied to a twodimensional incompressible viscous flow problem as a basic study for the fluid-structure interaction analysis using IGA. The vortex induced vibration of a circular cylinder is investigated as a numerical example, and the effectiveness and validity of the coupled analysis using IGA are discussed.

2-D local hp adaptive isogeometric analysis based on hierarchical Fup basis functions

The highly anisotropic, multi‐scale nature of fusion plasma simulations in tokamaks, combined with the complexity in the geometries of plasma‐facing components and magnetic equilibrium, challenges numerical schemes. They therefore require the development of advanced numerical techniques to enhance computational efficiency and enable codes to simulate realistic plasma configurations relevant to tokamak operation. This paper proposes an adaptive mesh refinement strategy (‐adaptivity) in the SolEdge‐HDG code for the resolution of 2D fluid‐drift Braginskii equations using the Hybrid Discontinuous Galerkin (HDG) method. The strategy is based on an oscillation indicator implemented to detect under‐resolved regions and dynamically refine the mesh, associated with an a posteriori accuracy indicator built on the local difference between the solution at order and the post‐processed one at order considered as reference. The method thus enables both refinement, where necessary, as well as coarsening in regions where the solution is smooth. Numerical results obtained with this method in realistic tokamak geometry and plasma conditions show significant reductions in computational resources and an improvement in code robustness while maintaining high accuracy, particularly in regions with steep gradients or near the sharp angles of the tokamak walls. This work highlights the potential of such ‐adaptivity technique to optimize transport simulations in realistic tokamak configurations, offering a fully automated, goal‐oriented mesh refinement strategy. In addition to optimizing the numerical cost of simulation, this strategy offers all users a fully automated means of designing a mesh in any tokamak geometry.

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

Adaptive isogeometric multi-material topology optimization based on suitably graded truncated hierarchical B-spline

In this paper we develop an a posteriori error analysis for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem. The governing system corresponds to a scalar, nonlinear convection-diffusion equation coupled with a Stokes problem with variable viscosity, and it serves as a prototype model for sedimentation-consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest. The solvability of the continuous mixed-primal formulation along with a priori error estimates for a finite element scheme using Raviart−Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., ESAIM: M2AN 49 (2015) 1399–1427]. Here we derive two efficient and reliable residual-based a posteriori error estimators for that scheme: for the first estimator, and under suitable assumptions on the domain, we apply a Helmholtz decomposition and exploit local approximation properties of the Clement interpolant and Raviart−Thomas operator to show its reliability. On the other hand, its efficiency follows from inverse inequalities and the localization arguments based on triangle-bubble and edge-bubble functions. Secondly, an alternative error estimator is proposed, whose reliability can be proved without resorting to Helmholtz decompositions. Our theoretical results are then illustrated via some numerical examples, highlighting also the performance of the scheme and properties of the proposed error indicators.