Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA) for the spatial discretization significantly reduces the pollution error. However, solving the resulting linear systems by means of a direct solver remains computationally expensive when large wave numbers or multiple dimensions are considered. An alternative lies in the use of (preconditioned) Krylov subspace methods. Recently, the use of the exact Complex Shifted Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to wave number independent convergence while obtaining more accurate numerical solutions using IgA. In this paper, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using a geometric multigrid method. Numerical results obtained for one- and two-dimensional model problems, including constant and non-constant wave numbers, show scalable convergence with respect to the wave number and approximation order p of the spatial discretization. Furthermore, when kh is kept constant, the proposed approach leads to a significant reduction of the computational time compared to the use of the multigrid-approximated or exact inverse of the CSLP with a small shift, in particular for three-dimensional model problems.@en

The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing. The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior. The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image.@en

The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing. The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior. The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image.@en

The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing. The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior. The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image.@en

The Material Point Method (MPM) is a numerical technique that combines a fixed Eulerian background grid and Lagrangian point masses to simulate materials which undergo large deformations. Within the original MPM, discontinuous gradients of the piecewise-linear basis functions lead to the so-called grid-crossing errors when particles cross element boundaries. Previous research has shown that B-spline MPM (BSMPM) is a viable alternative not only to MPM, but also to more advanced versions of the method that are designed to reduce the grid-crossing errors. In contrast to many other MPM-related methods, BSMPM has been used exclusively on structured rectangular domains, considerably limiting its range of applicability. In this paper, we present an extension of BSMPM to unstructured triangulations. The proposed approach combines MPM with C1-continuous high-order Powell–Sabin spline basis functions. Numerical results demonstrate the potential of these basis functions within MPM in terms of grid-crossing-error elimination and higher-order convergence.@en

The classical material point method (MPM) developed in the 90s is known for drawbacks which affect the quality of results. The movement of material points from one element to another leads to non-physical oscillations known as ‘grid crossing errors’. Furthermore, the use of material points as integration points renders a numerical quadrature rule of limited quality. Different solutions have been proposed in recent years to overcome these drawbacks. In this paper the approach of combining quadratic B-spline basis functions with a reconstruction based quadrature rule is pursued to solve these numerical problems. High-order B-spline basis functions solve the problem of grid crossing completely, whereas the considered reconstruction based quadrature rule reduces the quadrature error observed with MPM. In addition, the use of quadratic B-splines leads to a more accurate piecewise linear approximation of the stress field compared to the piecewise constant one obtained with linear Lagrangian basis functions commonly used with MPM. Two 1D benchmarks are considered involving large deformations, a vibrating bar and a column under self-weight. They render excellent results when adopting this high-order MPM.@en

Both the material-point method (MPM) and optimal transportation meshfree (OTM) method have been developed to efficiently solve partial differential equations that are based on the conservation laws from continuum mechanics. However, the methods are derived in a different fashion and have been studied independently of one another. In this paper, we provide a direct step-by-step comparison of the MPM and OTM algorithms. Based on this comparison, we derive the conditions, under which the two approaches can be related to each other, thereby bridging the gap between the MPM and OTM communities. In addition, we introduce a novel unified approach that combines the design principles from B-spline MPM and the OTM method. The proposed approach does not contain user-defined parameters and can decrease the costs of the standard OTM method. Moreover, it allows for the use of a consistent mass matrix without stability issues that are typically encountered in MPM computations.@en

The use of sequential time integration schemes becomes more and more the bottleneck within large-scale computations due to a stagnation of processor’s clock speeds. In this study, we combine the parallel-in-time Multigrid Reduction in Time method with a p-multigrid method to obtain a scalable solver specifically designed for Isogeometric Analysis. Numerical results obtained for two- and three-dimensional benchmark problems show the overall scalability of the proposed method on modern computer architectures and a significant improvement in terms of CPU timings compared to the use of standard spatial solvers.@en

Over the years, Isogeometric Analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p-multigrid method, in which coarsening is applied in the spline degree p instead of the mesh width h, and compare it to h-multigrid methods. Since the use of classical smoothers (e.g. Gauss–Seidel) results in a p-multigrid/h-multigrid method with deteriorating performance for higher values of p, the use of an ILUT smoother is investigated as well. Numerical results and a spectral analysis indicate that the use of this smoother exhibits convergence rates essentially independent of h and p for both p-multigrid and h-multigrid methods. In particular, we compare both coarsening strategies (e.g. coarsening in h or p) adopting both smoothers for a variety of two and three dimensional benchmarks. Furthermore, the ILUT smoother is compared to a state-of-the-art smoother (Hofreither and Takacs 2017) using both coarsening strategies. Finally, the proposed p-multigrid method is used to solve linear systems resulting from THB-spline discretizations.@en

Both the Material Point Method (MPM) and meshfree schemes based on optimal transport theory have been developed for efficient and robust integration of the weak form equations originating from computational mechanics. Although the methods are derived in a different fashion, their algorithms share many similarities. In this paper, we outline the close resemblance of MPM and Optimal Transportation Meshfree (OTM) schemes. Aside from a theoretical analysis, the methods are compared numerically using a one-dimensionalbenchmark.@en

Isogeometric Analysis is a methodology that bridges the gap between Computer Aided Design (CAD) and the Finite Element Method (FEM) by adopting the building blocks used in CAD, namely Non-UniformRational B-Splines and B-splines, as a basis for FEM. The use of these high-order spline functions does not only lead to an accurate representation of the geometry, but has shown to be advantageous in many different fields of research. In order to obtain accurate numerical solutions, sufficiently fine meshes have to be considered which results in very large linear systems of equations. Furthermore, the condition numbers of the system matrices grow exponentially in the spline degree p, making the use of standard iterative solvers less efficient. Direct methods, on the other hand, might not be a viable alternative for large problem sizes, due to memory constraints and difficulties to parallelize. In recent years, the development of efficient iterative solvers for Isogeometric Analysis has therefore become an active field of research. For standard FEM, multigrid methods are known to be among the most efficient solvers for elliptic partial differential equations. The direct application of these methods to linear systems arising in Isogeometric Analysis results, however, in multigrid methods with deteriorating performance for higher values of the spline degree p, since the multigrid smoother becomes less and less effective in damping the error. This has led to the development of multigrid methods with non-standard smoothers. In this dissertation,we propose the use of p-multigridmethods as an alternative solution strategy. Within our p-multigrid method, the coarse grid correction is obtained at level p Æ 1, enabling the use of well-known solution methods for standard Lagrangian FEM (in particular h-multigrid methods). Furthermore, the support of the basis functions significantly reduces at level p Æ 1, thereby reducing the number of non-zero entries in the coarse grid operators. We analyze the performance of our p-multigrid method, adopting different smoothers, for single patch and multipatch geometries. In particular, we perform a spectral analysis to investigate the interplay between the coarse grid correction and smoothing procedure and obtain the asymptotic convergence rate of the p-multigrid method for a representative scenario. Numerical results (i.e., iteration numbers and CPU timings) are obtained for a variety of two- and three-dimensional benchmarks and compared to (state-of-the-art) h-multigrid methods to show the potential of p-multigrid methods in the context of Isogeometric Analysis. For time-dependent partial differential equations, we apply Multigrid Reduced in Time (MGRIT), which is a parallel-in-time method, on discretizations arising in Isogeometric Analysis. Here, MGRIT is successfully combined with a p-multigrid method to obtain an overall efficient method. @en

Introduced in [1], Isogeometric Analysis (IgA) has become widely accepted in academia and industry. However, solving the resulting linear systems remains a challenging task. For instance, the condition number of the Poisson operator scales quadratically with the mesh width h, but, in contrast to standard Finite Elements, exponentially with the order of the approximation p [2]. The performance of (standard) iterative solvers thus decreases fast for higher values of p. In this talk we propose an efficient solution strategy for IgA discretizations that is based on p- multigrid techniques used both as a solver and as a preconditioner in a Krylov subspace iteration method. The approach makes use of a hierarchy of B-spline based discretizations of different approximation orders, which is in contrast to (geometric) h-multigrid methods, where a hierarchy of coarser and finer meshes is constructed. The `coarse grid' correction is determined at level p = 1, which enables us to use established solution techniques developed for low-order Lagrange finite elements. Prolongation and restriction operators are defined as mappings between arbitrary spline spaces, solely determined by the generating knot vectors, allowing us to combine coarsening in both h and p, leading to a flexible hp-multigrid. Prelimenary numerical results are presented for different two-dimensional benchmark problems on non-trivial geometries. It follows from a Local Fourier Analysis [3], that the coarse grid correction and the smoothing procedure complement each other quite well. Moreover, the obtained convergence rates indicate that p-multigrid methods have the potential to efficiently solve IgA discretizations. @en

The Material Point Method (MPM) has been applied successfully to problems in engineering which involve large deformations and history-dependent material behavior. However, the classical method suffers from some shortcomings which influence the quality of the numerical solution significantly. High-order B-spline basis functions solve the problem of so-called ‘grid crossing errors’ completely due to their higher continuity at inter-element boundaries. Adopting a consistent mass matrix instead of its lumped counterpart, which is common practice in standard MPM, further improves the convergence properties of the MPM. However, solving a linear system of equations resulting from a B-spline discretization is considered a challenging task. In this paper, we present a solution technique using p-multigrid methods to efficiently solve linear systems arising in B-spline MPM. @en

Isogeometric Analysis (IgA) can be seen as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. Combined with a time inte- gration scheme within the method of lines, IgA has become a viable alternative to FEM for time-dependent problems. However, as processors' clock speeds are no longer increasing but the number of cores are going up, traditional (i.e., sequential) time integration schemes become more and more the bottleneck within these large-scale computations. The Multigrid Reduced in Time (MGRIT) method is a parallel-in-time integration method that enables exploitation of parallelism not only in space but also in the temporal direction. In this paper, we apply MGRIT to discretizations arising from IgA for the _rst time in the literature. In particular, we investigate the (parallel) performance of MGRIT in this context for a variety of geometries, MGRIT hierarchies and time integration schemes. Numerical results show that the MGRIT method converges independent of the mesh width, spline degree of the B-spline basis functions and time step size _t and is highly parallelizable when applied in the context of IgA. @en

Isogeometric Analysis (IgA) can be considered as the natural extension of the Finite Element Method (FEM) to high-order B-spline basis functions. The development of efficient solvers for discretizations arising in IgA is a challenging task, as most (standard) iterative solvers have a detoriating performance for increasing values of the approximation order p of the basis functions. Recently, p-multigrid methods have been developed as an alternative solution strategy. With p-multigrid methods, a multigrid hierarchy is constructed based on the approximation order p instead of the mesh width h (i.e. h-multigrid). The coarse grid correction is then obtained at level p = 1, where B-spline basis functions coincide with standard Lagrangian P1 basis functions, enabling the use of well known solution strategies developed for the Finite Element Method to solve the residual equation. Different projection schemes can be adopted to go from the high-order level to level p = 1. In this paper, we compare a direct projection to level p = 1 with a projection between each level 1 ≤ k ≤ p in terms of iteration numbers and CPU times. Numerical results, including a spectral analysis, show that a direct projection leads to the most efficient method for both single patch and multipatch geometries.@en

Since its introduction in [20], Isogeometric Analysis (IgA) has established itself as a viable alternative to the Finite Element Method (FEM). Solving the resulting linear systems of equations efficiently remains, however, challenging when high-order B-spline basis functions of order p> 1 are adopted for approximation. The use of Incomplete LU (ILU) type factorizations, like ILU(k) or ILUT, as a preconditioner within a Krylov method or as a smoother within a multigrid method is very effective, but costly [37]. In this paper, we investigate the use of a block ILUT smoother within a p-multigrid method, where the coarse grid correction is obtained at p= 1, and compare it to a global ILUT smoother in case of multipatch geometries. A spectral analysis indicates that the use of the block ILUT smoother improves the overall convergence rate of the resulting p-multigrid method. Numerical results, obtained for a variety of two dimensional benchmark problems, illustrate the potential of this block ILUT smoother for multipatch geometries.@en

Isogeometric Analysis can be considered as the natural extension of the Finite Element Method (FEM) to higher-order spline based discretizations simplifying the treatment of complex geometries with curved boundaries. Finding a solution of the resulting linear systems of equations efficiently remains, however, a challenging task. Recently, p-multigrid methods have been considered [18], in which a multigrid hierarchy is constructed based on different approximation orders p instead of mesh widths h as it would be the case in classical h-multigrid schemes [8]. The use of an Incomplete LU-factorization as a smoother within the p-multigrid method has shown to lead to convergence rates independent of both h and p for single patch geometries [19]. In this paper, the focus lies on the application of the aforementioned p-multigrid method on multipatch geometries having a C0-continuous coupling between the patches. The use of ILUT as a smoother within p-multigrid methods leads to convergence rates that are essentially independent of h and p, but depend mildly on the number of patches.@en