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.

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The term smart grid is typically associated with power systems, but it can also be applied to gas and heat networks. There is no consensus on the definition of smart (gas) grids in the literature. We use the description provided in [27]: the smart gas grid concept is based on maximizing the efficiency of overall energy usage and taking full advantage of the exibility and all the opportunities that gas and the gas grid can offer. In this report we present the results of a six-months project on technology and policy aspects of smart gas grids, performed between October 2014 and April 2015. The project has led to a successful Energy System Integration: planning, operations, and societal embedding (ESI-pose) application in 2015, increasing the degree of collaboration between the faculty of Technology, Policy and Management (TPM) and Numerical Analysis section of Electrical Engineering, Mathematics and Computer Science (EEMSC) faculty. @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.

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The material-point method (MPM) is a continuum-based numerical tool to simulate problems that involve large deformations. Within MPM, a continuum is discretized by defining a set of Lagrangian particles, called material points, which store all relevant material properties. Themethod adopts an Eulerian background grid, where the equations of motion are solved at every time step. The solution on the background grid is used to subsequently update all material-point properties, such as displacement, velocity, and stress. In this way, MPM incorporates both Eulerian and Lagrangian descriptions. Similarly to other combined Eulerian-Lagrangian techniques, MPM attempts to avoid the numerical difficulties arising from nonlinear convective terms associated with an Eulerian problem formulation, while preventing grid distortion, typically encounteredwithin meshbased Lagrangian formulations. Over the years,MPM has been successfully applied to many complex problems from engineering and computer graphics. Despite its impressive performance for these applications, the method still suffers from several numerical shortcomings, such as stability issues, inaccurate mapping of the material-point data, and unphysical oscillations that arise when material points travel from one element to another, the so-called grid crossing errors. This dissertation provides an overview of the existing literature that addresses these drawbacks, and introduces new mathematical techniques that improve the performance of MPM. Previous studies have indicated that the use of higher-order B-spline basis functions within MPM mitigates the grid-crossing errors, thereby improving the accuracy of the method. This thesis combines the B-spline approach, known as BSMPM, with an alternative technique to project the information from material points to the background grid. The mapping technique is based on cubic-spline interpolation and Gauss quadrature. The numerical results show that the proposed approach further increases the accuracy of the method and leads to higher-order convergence. Moreover, the extension of BSMPM to unstructured grids using Powell-Sabin splines is discussed. After that, this dissertation compares MPM to the optimal transportation meshfree (OTM) method. Both MPM and the OTM method have been developed to efficiently solve partial differential equations that arise from the conservation laws in continuum mechanics. However, the methods are derived in a different fashion and have been studied independently of one another. This thesis provides a direct step-by-step comparison of the MPM and OTM algorithms. Based on this comparison, the conditions, under which the two approaches can be related to each other, are derived, thereby bridging the gap between the MPM and OTM communities. In addition, the thesis introduces a novel unified approach that combines the design principles from BSMPM and the OTM method. The proposed approach is significantly cheaper and more robust than the standard OTM method and allows for the use of a consistent mass matrix without stability issues that are typically encountered in MPM computations. Finally, this thesis introduces a novel function reconstruction technique that combines the well-known least-squares method with local Taylor basis functions, called Taylor least squares (TLS). The technique reconstructs functions from scattered data, while preserving their integral values. In conjunction with MPM or a related method, the TLS technique locally approximates quantities of interest, such as stress and density, and when used with a suitable quadrature rule, conserves the total mass and linear momentum after transferring the material-point information to the grid. The integration of the technique into MPM, dual domain material-point method (DDMPM), and BSMPM significantly improves the results of these methods. For the considered onedimensional examples, the TLS function reconstruction technique resembles the approximation properties of the highly-accurate cubic-spline reconstruction, while preserving the physical aspects of the standard algorithm. @en

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 the Taylor least squares (TLS), to reconstruct functions from scattered data while preserving their integrals. The TLS technique locally approximates quantities of interest such as stress and density, and when used with a suitable quadrature rule, it conserves the total mass and linear momentum after transferring the material-point information to the grid. The integration of the technique into MPM, dual domain MPM, and B-spline MPM significantly improves the results of these methods. For the considered examples, the TLS function reconstruction technique resembles the approximation properties of highly accurate spline reconstruction while preserving the physical properties of the standard algorithm.

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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

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 gridcrossing occurs more frequently, the convergence rate of B-spline MPMis lower. In this paper, moving least squares reconstruction is employed to retrieve the expected convergence rate. The proposed improvement is examined in terms of the spatial convergence using the methods of manufactured solutions for large deformations.@en

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 functions from scattered data. The TLS technique locally approximates quantities of interest, such as stress and density, and when used with a suitable quadrature rule, conserves the total mass and linear momentum after transferring the material-point information to the grid. For one-dimensional examples, applying the TLS approximation significantly improves the results of MPM, Dual Domain Material Point Method (DDMPM), and B-spline MPM (BSMPM). Due to its outstanding conservation properties, the TLS technique outperforms the nonconservative reconstruction techniques, such as spline reconstruction. For example, in contrast to the solution generated using the global cubic-spline interpolation, the TLS solution satisfies the boundary conditions of a two-phase benchmark. Therefore, the TLS reconstruction increases the accuracy of the material point methods, while preserving the fundamental physical properties of the standard algorithm. @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

Soil liquefaction describes a loss of strength of saturated sand upon sudden or cyclic loading. A slight disturbance of such a soil’s fabric might lead to severe damage, e.g. the collapse of sea dikes. Accurate modeling of the state transition between saturated soil and a liquefied soil-water mixture, as well as post-liquefaction phenomena, is crucial for the prediction of such damage. However, developing an appropriate numerical model remains a challenging problem, especially when the simulation involves dynamic large deformation processes. In order to make a first step towards an accurate simulation of soil liquefaction, a two-phase formulation of the finite element method (FEM) in conjunction with the elastoplastic UBC3D-PLM model is investigated. The performance of this approach is analyzed based on a shaking table benchmark.@en

Commercial reservoir simulators must be very robust and fast. Moreover, current hardware requires the simulators to scale over multiple number of computing nodes and for a fixed (‘strong scalability’) as well as an increasing problem size per computing node (‘weak scalability’). In most current commercial reservoir simulators, due to the different geological structures and properties of hydrocarbon reservoirs and the use of enhanced oil recovery (EOR) techniques, the governing equations are strongly nonlinear and hard to solve. The Jacobian system is solved by FGMRES preconditioned by the two-level constrained pressure residual (CPR) preconditioner. The driving force of the CPR preconditioner is the solution of the pressure equation. The industry standard for solving the pressure equation is the algebraic multigrid (AMG) solver. AMG is well known for its ‘weak scalability’. However, in these applications, AMG has unfavorable ‘strong’ scalability properties. This degradation in scalability is due to the increased level of inter-processor communication in the algorithm. In this paper, a monotone non-Galerkin AMG (MNG-AMG) method is presented. The aim of the method is to reduce the overall communication in MNG-AMG by enforcing a predefined nonzero pattern and monotonicity property (i.e., M-matrices) on each multigrid level. This paper describes the application of the MNG-AMG method in the context of reservoir simulations. We will compare the parallel scalability of the default solver with the MNG-AMG solver and discuss the optimal values for the MNG-AMG solver for a variety of test cases based on full field reservoir simulations.@en