Combining p-multigrid and Multigrid Reduction in Time methods to obtain a scalable solver for Isogeometric Analysis

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...

Multigrid Reduced in Time for Isogeometric Analysis

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...

A Block ILUT Smoother for Multipatch Geometries in Isogeometric Analysis

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...

P-multigrid methods for Isogeometric analysis

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...

Towards accuracy and scalability: Combining Isogeometric Analysis with deflation to obtain scalable convergence for the Helmholtz equation

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)...

A Direct Projection to Low-Order Level for p-Multigrid Methods in Isogeometric Analysis

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...

Efficient p-Multigrid Based Solvers for Isogeometric Analysis on Multipatch Geometries

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...

Comparison and unification of material-point and optimal transportation meshfree methods

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...

Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

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...

p-multigrid methods and their comparison to h-multigrid methods within Isogeometric Analysis

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...

Efficient multigrid based solvers for B-spline MPM

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...

Comparison between Material Point Method and meshfree schemes derived from optimal transportation theory

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...

Recent Developments to Improve the Numerical Accuracy

Efficient Multigrid based solvers for isogeometric analysis

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...

A High Order Material Point Method

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...