Matthias Möller

Delft University of Technology | TU · Department of Applied Mathematics

Accelerating topology optimization using efficient algorithms on GPU hardware.

Near term quantum devices have the potential to outperform classical computing through the use of hybrid classical-quantum algorithms such as Variational Quantum Eigensolvers. These iterative algorithms use a classical optimiser to update a parameterised quantum circuit. Each iteration, the circuit is executed on a physical quantum processor or qua...

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....

This paper presents the benchmark score definitions of QPack, an application-oriented cross-platform benchmarking suite for quantum computers and simulators, which makes use of scalable Quantum Approximate Optimization Algorithm and Variational Quantum Eigensolver applications. Using a varied set of benchmark applications, an insight of how well a...

Triangulated meshes discretized from commercial CAD applications often possess a considerable level of complexity. However, when conducting external aerodynamics simulations at an earlier design stage, these meshes are way too complex and contain complex features and topological holes. We propose a practical and fast algorithm to shrink wrap triang...

Using the standard finite element method (FEM) to solve general partial differential equations, the round-off error is found to be proportional to $N^{\beta_{\rm R}}$, with $N$ the number of degrees of freedom (DoFs) and $\beta_{\rm R}$ a coefficient. A method which uses a few cheap numerical experiments is proposed to determine the coefficient of...

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 (...

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....

Isogeometric Analysis (IgA) has become a viable alternative to the Finite Element Method (FEM) and is typically combined with a time integration scheme within the method of lines for time-dependent problems. However, due to a stagnation of processors clock speeds, traditional (i.e. sequential) time integration schemes become more and more the bottl...

Isogeometric Analysis [1] has become increasingly popular as an alternative to the Finite Element Method. Solving the resulting linear systems when adopting higher order B-spline basis functions remains a challenging task, as most (standard) iterative methods have a deteriorating preformance for higher values of the approximation order p.Recently,...

Modelling nonlinear phenomena in thin rubber shells calls for stretch-based material models, such as the Ogden model which conveniently utilizes eigenvalues of the deformation tensor. Derivation and implementation of such models have been already made in Finite Element Methods. This is, however, still lacking in shell formulations based on Isogeome...

This paper proposes a shape optimization algorithm based on the principles of Isogeometric Analysis (IGA) in which the parameterization of the geometry enters the problem formulation as an additional PDE-constraint. Inspired by the isoparametric principle of IGA, the parameterization and the governing state equation are treated using the same numer...

In finite element methods, the accuracy of the solution cannot increase indefinitely since the round-off error related to limited computer precision increases when the number of degrees of freedom (DoFs) is large enough. Because a priori information of the highest attainable accuracy is of great interest, we construct an innovative method to obtain...

In this paper, we present QPack, a benchmark for NISQ era quantum computers using QAOA algorithms. Unlike other evaluation metrics in the field, this benchmark evaluates not only one, but multiple important aspects of quantum computing hardware: the maximum problem size a quantum computer can solve, the required run-time, as well as the achieved ac...

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...

The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from no more than a description of the boundary contours of the physical domain. A subclass of the multitude of the...

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-multigr...

This proceedings volume gathers a selection of outstanding research papers presented at the third Conference on Isogeometric Analysis and Applications, held in Delft, The Netherlands, in April 2018. This conference series, previously held in Linz, Austria, in 2012 and Annweiler am Trifels, Germany, in 2014, has created an international forum for in...

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...

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...

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 approximatio...

Wrinkling or pattern formation of thin (floating) membranes is a phenomenon governed by buckling instabilities of the membrane. For (post-) buckling analysis, arc-length or continuation methods are often used with a priori applied perturbations in order to avoid passing bifurcation points when traversing the equilibrium paths. The shape and magnitu...

The Hodgkin-Huxley (HH) neuron is one of the most biophysically-meaningful models used in computational neuroscience today. Ironically, the model’s high experimental value is offset by its disproportional computational complexity. To such an extent that neuroscientists have either resorted to simpler models, losing precious neuron detail, or to usi...

This paper introduces a new cross-platform programming framework for developing quantum-accelerated scientific computing applications and executing them on most of today’s cloud-based quantum computers and simulators. It makes use of C++ template meta-programming techniques to implement quantum algorithms as generic, platform-independent expression...

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 b...

In this work, we consider a Cahn–Hilliard phase field-based computational model for immiscible and incompressible two-component liquid flows with interfacial phenomena. This diffuse-interface complex-fluid model is given by the incompressible Navier–Stokes–Cahn–Hilliard (NSCH) equations. The coupling of the flow and phase field equations is given b...

Isogeometric analysis was applied very successfully to many problem classes like linear elasticity, heat transfer and incompressible flow problems but its application to compressible flows is very rare. However, its ability to accurately represent complex geometries used in industrial applications makes IGA a suitable tool for the analysis of compr...

This work extends the high-resolution isogeometric analysis approach established in chapter “High-Order Isogeometric Methods for Compressible Flows. I: Scalar Conservation Laws” (Jaeschke and Möller: High-order isogeometric methods for compressible flows. I. Scalar conservation Laws. In: Proceedings of the 19th International Conference on Finite El...

Computational neuroscience uses models to study the brain. The Hodgkin-Huxley (HH) model, and its extensions, is one of the most powerful, biophysically meaningful models currently used. The high experimental value of the (extended) Hodgkin-Huxley (eHH) models comes at the cost of steep computational requirements. Consequently, for larger networks,...

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...

This paper proposes a shape optimization algorithm based on the principles of Isogeometric Analysis (IGA) in which the parameterization of the geometry enters the problem formulation as an additional PDE-constraint. Inspired by the isoparametric principle of IGA, the parameterization and the governing state equation are treated using the same numer...

This paper presents a PDE-based planar parameterization framework with support for Truncated Hierarchical B-Splines (THB-splines). For this, we adopt the a posteriori refinement strategy of Dual Weighted Residual and present several adaptive numerical schemes for the purpose of approximating an inversely harmonic geometry parameterization. Hereby,...

In finite element methods (FEMs), the accuracy of the solution cannot increase indefinitely because the round-off error increases when the number of degrees of freedom (DoFs) is large enough. This means that the accuracy that can be reached is limited. A priori information of the highest attainable accuracy is therefore of great interest. In this p...

This paper presents a novel spline-based meshing technique that allows for usage of boundary-conforming meshes for unsteady flow and temperature simulations in co-rotating twin-screw extruders. Spline-based descriptions of arbitrary screw geometries are generated using Elliptic Grid Generation. They are evaluated in a number of discrete points to y...

We propose a numerical scheme based on the principles of Isogeometric Analysis (IgA) for a geometrical pattern formation induced evolution of manifolds. The development is modelled by the use of the Gray-Scott equations for pattern formation in combination with an equation for the displacement of the manifold. The method forms an alternative to the...

This paper presents a novel spline-based meshing technique that allows for usage of boundary-conforming meshes for unsteady flow and temperature simulations in co-rotating twin-screw extruders. Spline-based descriptions of arbitrary screw geometries are generated using Elliptic Grid Generation. They are evaluated in a number of discrete points to y...

The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from no more than a description of the boundary contours of the physical domain. A subclass of the multitude of the...

Within the original Material Point Method (MPM), discontinuous gradients of the piece-wise linear basis functions lead to so-called `grid-crossing errors' when particles cross element boundaries. This can be overcome by using $C^1$-continuous basis functions such as higher-order B-splines. In this paper, we extend this approach to unstructured tria...

The development of practical quantum computers that can be used to solve real-world problems is in full swing driven by the ambitious expectation that quantum supremacy will be able to outperform classical super-computers. Like with any emerging compute technology, it needs early adopters in the scientific computing community to identify problems o...

The paper shows a moving least squares reconstruction technique applied to the B-spline Material Point Method (B-spline MPM). It has been shown previously that B-spline MPM can reduce grid-crossing errors inherent in the original Material Point Method. However, in the large deformation regime where the grid crossing occurs more frequently, the conv...

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 each level of the hierarchy is associated with a different approximation order...

The fully automated generation of computational meshes for twin-screw machine geometries constitutes a mandatory aspect for the numerical simulation (and shape-optimization) of these devices but proves to be a challenging task in practice. Therefore, the successful generation of computational meshes requires sophisticated mathematical tools. Commer...

This paper reports on the current status of an isogeometric modeling and analysis framework for rotary twin-screw machines that is being developed by an international consortium of academic partners within the EU-funded MOTOR project. The approach aims at combining accurate geometry modeling capabilities with modern high-performance computing techn...

Isogeometric analysis was applied very successfully to many problem classes like linear elasticity, heat transfer and incompressible flow problems but its application to compressible flows is very rare. However, its ability to accurately represent complex geometries used in industrial applications makes IGA a suitable tool for the analysis of compr...

This work extends the high-resolution isogeometric analysis approach established for scalar transport equations to the equations of gas dynamics. The group finite element formulation is adopted to obtain an efficient assembly procedure for the standard Galerkin approximation, which is stabilized by adding artificial viscosities proportional to the...

High-performance computing platforms are becoming more and more heterogeneous, which makes it very difficult for researchers and scientific software developers to keep up with the rapid changes on the hardware market. In this paper, the open-source project FDBB (Fluid Dynamics Building Blocks) is presented, which eases the development of fluid dyna...

Within the standard Material Point Method (MPM), the spatial errors are partially caused by the direct mapping of material‐point data to the background grid. In order to reduce these errors, we introduced a novel technique that combines the Least Squares method with the Taylor basis functions, called Taylor Least Squares (TLS), to reconstruct funct...

A generic particle-mesh method using a hybridized discontinuous Galerkin (HDG) framework is presented and validated for the solution of the incompressible Navier-Stokes equations. Building upon particle-in-cell concepts, the method is formulated in terms of an operator splitting technique in which Lagrangian particles are used to discretize an adve...

The generation of an analysis-suitable computational grid from a description of no more than its boundaries is a common problem in numerical analysis. Most classical meshing techniques for finite-volume, finite-difference or finite-element applications such as the Advancing Front Method (Schöberl, 1997), Delaunay Triangulation (Triangle, 1996) and...

In this work the feasibility of a numerical wave tank using a hybrid particle-mesh method is investigated. Based on the Fluid Implicit Particle Method (FLIP) a generic formulation for the hybrid method is presented for incompressible multi-phase flows involving large density jumps and wave generating boundaries. The performance of the method is ass...

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 o...

Quantum computing technologies have become a hot topic in academia and industry receiving much attention and financial support from all sides. Building a quantum computer that can be used practically is in itself an outstanding challenge that has become the 'new race to the moon'. Next to researchers and vendors of future computing technologies, na...

This poster presents the outline of a high-order particle-mesh method for the incompressible Navier-Stokes equations.

Ordinary Differential Equations (ODEs) are widely used in many high-performance computing applications. However, contemporary processors generally provide limited throughput for these kinds of calculations. A high-performance hardware accelerator has been developed for speeding-up the solution of ODEs. The hardware accelerator has been developed bo...

In this work the feasibility of a numerical wave tank using a hybrid particle-mesh method is investigated. Based on the fluid implicit particle method (FLIP) a formulation for the hybrid method is presented for incompressible multiphase flows involving large density jumps and wave generating boundaries. The performance of the method is assessed for...

In this work, we present our numerical results of the application of Galerkin-based Isogeometric Analysis (IGA) to incompressible Navier--Stokes--Cahn--Hilliard (NSCH) equations in velocity--pressure--phase field--chemical potential formulation. For the approximation of the velocity and pressure fields, LBB compatible non-uniform rational B-spline...

In this paper, a three-dimensional semi-idealized model for tidal motion in a tidal estuary of arbitrary shape and bathymetry is presented. This model aims at bridging the gap between idealized and complex models. The vertical profiles of the velocities are obtained analytically in terms of the first-order and the second-order partial derivatives o...

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity-pressure formulation of the steady state as well as the unsteady incompressible Navier-Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth genera...

Finite element methods are one of the most prominent discretisation techniques for the solution of partial differential equations. They provide high geometric flexibility, accuracy and robustness, and a rich body of theory exists. In this chapter, we summarise the main principles of Galerkin finite element methods, and identify and discuss avenues...

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications...

The algebraic flux-correction (AFC) approach introduced in [8, 17] for the accurate treatment of convection-dominated flow problems and refined in a series of publications [9,11-15,18,19] is extended to non-conforming finite element discretizations. Originally, this class of multidimensional high-resolution schemes was developed in the framework of...

Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity opera...

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields...

The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual...

The numerical treatment of flow problems by the finite element method is addressed. An algebraic approach to constructing high-resolution schemes for scalar conservation laws as well as for the compressible Euler equations is pursued. Starting from the standard Galerkin approximation, a diffusive low-order discretization is constructed by performin...

Grid adaptation based on the red-green strategy [1] is revisited. A new approach to the construction of conforming hybrid triangulations with linear time and space requirement is presented. The main advantage of the method is that mesh refinement and re-coarsening yields a sequence of hierarchical grids that can be used within multigrid schemes. Al...

A new generalization of the flux-corrected transport (FCT) methodology to implicit finite element discretizations is proposed. The underlying high-order scheme is supposed to be unconditionally stable and produce time-accurate solutions to evolutionary convection problems. Its nonoscillatory low-order counterpart is constructed by means of mass lum...

A discrete Newton approach is applied to implicit flux-limiting schemes based on the concept of algebraic flux correction. The Jacobian matrix is approximated by divided differences and assembled edge by edge. The use of a nodal flux limiter leads to an extended stencil which can be constructed a priori. Numerical examples for 2D benchmark problems...

The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time-stepping schemes. It is shown that Newton-like techniques can be applied to the nonlinear systems of equations resulting from the application of high-resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by mean...

New a posteriori error indicators based on edgewise slope-limiting are presented. The L2-norm is employed to measure the error of the solution gradient in both global and element sense. A second-order Newton–Cotes formula is utilized in order to decompose the local gradient error from a 1 finite element solution into a sum of edge contributions. Th...

Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edge-by-edge matrix assembly. A generalization of Roe’s approximate Riemann solver is derive...

An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This...

A slope limiting approach to the design of recovery based a posteriori error indicators for P 1 finite element discretizations is presented. The smoothed gradient field is recovered at edge midpoints by means of limited averaging of adjacent slope values. As an alternative, the constant gradient values may act as upper and lower bounds to be impose...

The flux-corrected transport (FCT) methodology is generalized to implicit finite element schemes and applied to the Euler equations of gas dynamics. For scalar equations, a local extremum diminishing scheme is constructed by adding artificial diffusion so as to eliminate negative off-diagonal entries from the high-order transport operator. To obtai...

The flux-corrected-transport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory low-order method of upwind type is derived by elimination of negative off-diagonal entries of the discrete transport operator. The difference between the discretizations...

The flux-corrected transport (FCT) methodology is generalized to implicit finite element schemes and applied to the Euler equations of gas dynamics. The underlying low-order scheme is constructed by applying scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. All conservative matrix manipulations are perfor...

The flux-corrected-transport paradigm is generalized to finite-element schemes based on arbitrary time stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of ne...

Even today, the accurate treatment of convection-dominated transport problems remains a challenging task in numerical simulation of both compressible and incompressible flows. The discrepancy arises between high accuracy and good resolution of singularities on the one hand and preventing the growth and birth of nonphysical oscillations on the oth...