arXiv subjects

Rolf Krause

Publications and source records attributed to Rolf Krause.

At least 73 records · Page 4Linked to original sources

arXiv2020

Frontiers in Mortar Methods for Isogeometric Analysis

Complex geometries as common in industrial applications consist of multiple patches, if spline based parametrizations are used. The requirements for the generation of analysis-suitable models are increasing dramatically since isogeometric analysis is directly based on the spline parametrization and nowadays used for the calculation of higher-order partial differential equations. The computational, or more general, the engineering analysis necessitates suitable coupling techniques between the different patches. Mortar methods have been successfully applied for coupling of patches and for contact mechanics in recent years to resolve the arising issues within the interface. We present here current achievements in the design of mortar technologies in isogeometric analysis within the Priority Program SPP 1748, Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretisation Methods, Mechanical and Mathematical Analysis.

cs.CE↗

arXiv2020

PIEMAP: Personalized Inverse Eikonal Model from cardiac Electro-Anatomical Maps

Electroanatomical mapping, a keystone diagnostic tool in cardiac electrophysiology studies, can provide high-density maps of the local electric properties of the tissue. It is therefore tempting to use such data to better individualize current patient-specific models of the heart through a data assimilation procedure and to extract potentially insightful information such as conduction properties. Parameter identification for state-of-the-art cardiac models is however a challenging task. In this work, we introduce a novel inverse problem for inferring the anisotropic structure of the conductivity tensor, that is fiber orientation and conduction velocity along and across fibers, of an eikonal model for cardiac activation. The proposed method, named PIEMAP, performed robustly with synthetic data and showed promising results with clinical data. These results suggest that PIEMAP could be a useful supplement in future clinical workflows of personalized therapies.

math.OC↗

arXiv2020

Large scale simulation of pressure induced phase-field fracture propagation using Utopia

Non-linear phase field models are increasingly used for the simulation of fracture propagation models. The numerical simulation of fracture networks of realistic size requires the efficient parallel solution of large coupled non-linear systems. Although in principle efficient iterative multi-level methods for these types of problems are available, they are not widely used in practice due to the complexity of their parallel implementation. Here, we present Utopia, which is an open-source C++ library for parallel non-linear multilevel solution strategies. Utopia provides the advantages of high-level programming interfaces while at the same time a framework to access low-level data-structures without breaking code encapsulation. Complex numerical procedures can be expressed with few lines of code, and evaluated by different implementations, libraries, or computing hardware. In this paper, we investigate the parallel performance of our implementation of the recursive multilevel trust-region (RMTR) method based on the Utopia library. RMTR is a globally convergent multilevel solution strategy designed to solve non-convex constrained minimization problems. In particular, we solve pressure-induced phase-field fracture propagation in large and complex fracture networks. Solving such problems is deemed challenging even for a few fractures, however, here we are considering networks of realistic size with up to 1000 fractures.

physics.comp-ph↗

arXiv2020

A Multilevel Approach to Training

We propose a novel training method based on nonlinear multilevel minimization techniques, commonly used for solving discretized large scale partial differential equations. Our multilevel training method constructs a multilevel hierarchy by reducing the number of samples. The training of the original model is then enhanced by internally training surrogate models constructed with fewer samples. We construct the surrogate models using first-order consistency approach. This gives rise to surrogate models, whose gradients are stochastic estimators of the full gradient, but with reduced variance compared to standard stochastic gradient estimators. We illustrate the convergence behavior of the proposed multilevel method to machine learning applications based on logistic regression. A comparison with subsampled Newton's and variance reduction methods demonstrate the efficiency of our multilevel method.

cs.LG↗

arXiv2020

Multilevel Minimization for Deep Residual Networks

We present a new multilevel minimization framework for the training of deep residual networks (ResNets), which has the potential to significantly reduce training time and effort. Our framework is based on the dynamical system's viewpoint, which formulates a ResNet as the discretization of an initial value problem. The training process is then formulated as a time-dependent optimal control problem, which we discretize using different time-discretization parameters, eventually generating multilevel-hierarchy of auxiliary networks with different resolutions. The training of the original ResNet is then enhanced by training the auxiliary networks with reduced resolutions. By design, our framework is conveniently independent of the choice of the training strategy chosen on each level of the multilevel hierarchy. By means of numerical examples, we analyze the convergence behavior of the proposed method and demonstrate its robustness. For our examples we employ a multilevel gradient-based methods. Comparisons with standard single level methods show a speedup of more than factor three while achieving the same validation accuracy.

cs.LG↗

arXiv2020

Modelling of hydro-mechanical processes in heterogeneous fracture intersections using a fictitious domain method with variational transfer operators

Fluid flow in rough fractures and the coupling with the mechanical behaviour of the fractures pose great difficulties for numerical modeling approaches, due to complex fracture surface topographies, the non-linearity of hydromechanical processes and their tightly coupled nature. To this end, we have adapted a fictitious domain method to enable the simulation of hydromechanical processes in fracture-intersections. The main characteristic of the method is the immersion of the fracture, modelled as a linear elastic solid, in the surrounding computational fluid domain, modelled with the incompressible Navier Stokes equations. The fluid and the solid problems are coupled with variational transfer operators. Variational transfer operators are also used to solve contact within the fracture using a dual mortar approach and to generate problem specific fluid meshes. With respect to our applications, the key features of the method are the usage of different finite element discretizations for the solid and the fluid problem and the automatically generated representation of the fluid-solid boundary. We demonstrate that the presented methodology resolves small-scale roughness on the fracture surface, while capturing fluid flow field changes during mechanical loading. Starting with 2D/3D benchmark simulations of intersected fractures, we end with an intersected fracture composed of complex fracture surface topographies, which are in contact under increasing loads. The contributions of this article are: (1) the application of the fictitious domain method to study flow in fractures with intersections, (2) a mortar based contact solver for the solid problem, (3) generation of problem specific grids using the geometry information from the variational transfer operators.

physics.comp-ph↗

arXiv2019

A Multigrid Method for a Nitsche-based Extended Finite Element Method

We present a tailored multigrid method for linear problems stemming from a Nitsche-based extended finite element method (XFEM). Our multigrid method is robust with respect to highly varying coefficients and the number of interfaces in a domain. It shows level independent convergence rates when applied to different variants of Nitsche's method. Generally, multigrid methods require a hierarchy of finite element (FE) spaces which can be created geometrically using a hierarchy of nested meshes. However, in the XFEM framework, standard multigrid methods might demonstrate poor convergence properties if the hierarchy of FE spaces employed is not nested. We design a prolongation operator for the multigrid method in such a way that it can accommodate the discontinuities across the interfaces in the XFEM framework and recursively induces a nested FE space hierarchy. The prolongation operator is constructed using so-called pseudo-$L^2$-projections; as common, the adjoint of the prolongation operator is employed as the restriction operator. The stabilization parameter in Nitsche's method plays an important role in imposing interface conditions and also affects the condition number of the linear systems. We discuss the requirements on the stabilization parameter to ensure coercivity and review selected strategies from the literature which are used to implicitly estimate the stabilization parameter. Eventually, we compare the impact of different variations of Nitsche's method on discretization errors and condition number of the linear systems. We demonstrate the robustness of our multigrid method with respect to varying coefficients and the number of interfaces and compare it with other preconditioners.

math.NA↗

arXiv2019

Contact between rough rock surfaces using a dual mortar method

The mechanical behavior of fractures in solids, such as rocks, has strong implications for reservoir engineering applications. Deformations, and the corresponding change in solid contact area and aperture field, impact rock fracture stiffness and permeability thus altering the reservoir properties significantly. Simulating contact between fractures is numerically difficult as the non-penetration constraints lead to a nonlinear problem and the surface meshes of the solid bodies on the opposing fracture sides may be non-matching. Furthermore, the challenging geometry of the arising constraints requires to solve the problem in several iterations, adjusting the constraints in each one. Here we present a novel discrete implementation of a dual mortar method and a non-smooth SQP solver, suitable for parallel computing, and apply it to a two body contact problem consisting of realistic rock fracture geometries from the Grimsel underground laboratory in Switzerland. The contributions of this article are: 1) a novel, parallel implementation of a dual mortar method and non-smooth SQP method, 2) realistic rock geometries with rough surfaces, and 3) numerical examples, which prove that the dual mortar method is capable of replicating the nonlinear closure behavior of fractures, observed in laboratory experiments.

physics.geo-ph↗

arXiv2019

Simulation of hydro-mechanically coupled processes in rough rock fractures using an immersed boundary method and variational transfer operators

Hydro-mechanical processes in rough fractures are highly non-linear and govern productivity and associated risks in a wide range of reservoir engineering problems. To enable high-resolution simulations of hydro-mechanical processes in fractures, we present an adaptation of an immersed boundary method to compute fluid flow between rough fracture surfaces. The solid domain is immersed into the fluid domain and both domains are coupled by means of variational volumetric transfer operators. The transfer operators implicitly resolve the boundary between the solid and the fluid, which simplifies the setup of fracture simulations with complex surfaces. It is possible to choose different formulations and discretization schemes for each subproblem and it is not necessary to remesh the fluid grid. We use benchmark problems and real fracture geometries to demonstrate the following capabilities of the presented approach: (1) Resolving the boundary of the rough fracture surface in the fluid; (2) Capturing fluid flow field changes in a fracture which closes under increasing normal load; and (3) Simulate the opening of a fracture due to increased fluid pressure.

physics.geo-ph↗

arXiv2019

Recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture

The simulation of crack initiation and propagation in an elastic material is difficult, as crack paths with complex topologies have to be resolved. Phase-field approach allows to simulate crack behavior by circumventing the need to explicitly model crack paths. However, the underlying mathematical model gives rise to a non-convex constrained minimization problem. In this work, we propose a recursive multilevel trust region (RMTR) method to efficiently solve such a minimization problem. The RMTR method combines the global convergence property of the trust region method and the optimality of the multilevel method. The solution process is accelerated by employing level dependent objective functions, minimization of which provides correction to the original/fine-level problem. In the context of the phase-field fracture approach, it is challenging to design efficient level dependent objective functions as the underlying mathematical model relies on the mesh dependent parameters. We introduce level dependent objective functions that combine fine level description of the crack path with the coarse level discretization. The overall performance and the convergence properties of the proposed RMTR method are investigated by means of several numerical examples in three dimensions.

math.NA↗

arXiv2019

High-dimensional and higher-order multifidelity Monte Carlo estimators

Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the accuracy of the estimates. This approach has recently gained widespread attention in uncertainty quantification. This is partly due to the availability of optimal strategies for the estimation of the expectation of scalar quantities-of-interest. In practice, the optimal strategy for the expectation is also used for the estimation of variance and sensitivity indices. However, a general strategy is still lacking for vector-valued problems, nonlinearly statistically-dependent models, and estimators for which a closed-form expression of the error is unavailable. The focus of the present work is to generalize the standard multifidelity estimators to the above cases. The proposed generalized estimators lead to an optimization problem that can be solved analytically and whose coefficients can be estimated numerically with few runs of the high- and low-fidelity models. We analyze the performance of the proposed approach on a selected number of experiments, with a particular focus on cardiac electrophysiology, where a hierarchy of physics-based low-fidelity models is readily available.

math.NA↗

arXiv2019

3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers

This work presents a modeling approach for single-phase flow in 3D fractured porous media with non-conforming meshes. To this end, a Lagrange multiplier method is combined with a parallel $L^2$-projection variational transfer approach. This Lagrange multiplier method enables the use of non-conforming meshes and depicts the variable coupling between fracture and matrix domain. The $L^2$-projection variational transfer allows general, accurate, and parallel projection of variables between non-conforming meshes (i.e. between fracture and matrix domain). Comparisons of simulations with 2D benchmarks show good agreement, and the method is further validated on 3D fracture networks by comparing it to results from conforming mesh simulations which were used as a reference. Application to realistic fracture networks with hundreds of fractures is demonstrated. Mesh size and mesh convergence are investigated for benchmark cases and 3D fracture network applications. Results demonstrate that the Lagrange multiplier method, in combination with the $L^2$-projection method, is capable of modeling single-phase flow through realistic 3D fracture networks.

physics.geo-ph↗

arXiv2018

A shortest-path based clustering algorithm for joint human-machine analysis of complex datasets

Clustering is a technique for the analysis of datasets obtained by empirical studies in several disciplines with a major application for biomedical research. Essentially, clustering algorithms are executed by machines aiming at finding groups of related points in a dataset. However, the result of grouping depends on both metrics for point-to-point similarity and rules for point-to-group association. Indeed, non-appropriate metrics and rules can lead to undesirable clustering artifacts. This is especially relevant for datasets, where groups with heterogeneous structures co-exist. In this work, we propose an algorithm that achieves clustering by exploring the paths between points. This allows both, to evaluate the properties of the path (such as gaps, density variations, etc.), and expressing the preference for certain paths. Moreover, our algorithm supports the integration of existing knowledge about admissible and non-admissible clusters by training a path classifier. We demonstrate the accuracy of the proposed method on challenging datasets including points from synthetic shapes in publicly available benchmarks and microscopy data.

q-bio.QM↗

arXiv2018

An immersed boundary method for fluid-structure interaction based on overlapping domain decomposition

We present a novel framework inspired by the Immersed Boundary Method for predicting the fluid-structure interaction of complex structures immersed in flows with moderate to high Reynolds numbers. The main novelties of the proposed fluid-structure interaction framework are 1) the use of elastodynamics equations for the structure, 2) the use of a high-order Navier-Stokes solver for the flow, and 3) the variational transfer (L2-projection) for coupling the solid and fluid subproblems. The dynamic behavior of a deformable structure is simulated in a finite element framework by adopting a fully implicit scheme for its temporal integration. It allows for mechanical constitutive laws including nonhomogeneous and fiber-reinforced materials. The Navier-Stokes equations for the incompressible flow are discretized with high-order finite differences which allow for the direct numerical simulation of laminar, transitional and turbulent flows. The structure and the flow solvers are coupled by using an L2-projection method for the transfer of velocities and forces between the fluid grid and the solid mesh. This strategy allows for the numerical solution of coupled large scale problems based on nonconforming structured and unstructured grids. The framework is validated with the Turek-Hron benchmark and a newly proposed benchmark modelling the flow-induced oscillation of an inert plate. A three-dimensional simulation of an elastic beam in transitional flow is provided to show the solver's capability of coping with anisotropic elastic structures immersed in complex fluid flow.

physics.comp-ph↗

arXiv2018

Smart energy models for atomistic simulations using a DFT-driven multifidelity approach

The reliability of atomistic simulations depends on the quality of the underlying energy models providing the source of physical information, for instance for the calculation of migration barriers in atomistic Kinetic Monte Carlo simulations. Accurate (high-fidelity) methods are often available, but since they are usually computationally expensive, they must be replaced by less accurate (low-fidelity) models that introduce some degrees of approximation. Machine-learning techniques such as artificial neural networks are usually employed to work around this limitation and extract the needed parameters from large databases of high-fidelity data, but the latter are often computationally expensive to produce. This work introduces an alternative method based on the multifidelity approach, where correlations between high-fidelity and low-fidelity outputs are exploited to make an educated guess of the high-fidelity outcome based only on quick low-fidelity estimations, hence without the need of running full expensive high-fidelity calculations. With respect to neural networks, this approach is expected to require less training data because of the lower amount of fitting parameters involved. The method is tested on the prediction of ab initio formation and migration energies of vacancy diffusion in iron-copper alloys, and compared with the neural networks trained on the same database.

physics.comp-ph↗

arXiv2018

Multiscale modeling, discretization, and algorithms: a survey in biomechanics

Multiscale models allow for the treatment of complex phenomena involving different scales, such as remodeling and growth of tissues, muscular activation, and cardiac electrophysiology. Numerous numerical approaches have been developed to simulate multiscale problems. However, compared to the well-established methods for classical problems, many questions have yet to be answered. Here, we give an overview of existing models and methods, with particular emphasis on mechanical and bio-mechanical applications. Moreover, we discuss state-of-the-art techniques for multilevel and multifidelity uncertainty quantification. In particular, we focus on the similarities that can be found across multiscale models, discretizations, solvers, and statistical methods for uncertainty quantification. Similarly to the current trend of removing the segregation between discretizations and solution methods in scientific computing, we anticipate that the future of multiscale simulation will provide a closer interaction with also the models and the statistical methods. This will yield better strategies for transferring the information across different scales and for a more seamless transition in selecting and adapting the level of details in the models. Finally, we note that machine learning and Bayesian techniques have shown a promising capability to capture complex model dependencies and enrich the results with statistical information; therefore, they can complement traditional physics-based and numerical analysis approaches.

math.NA↗

arXiv2018

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.

cs.CE↗

arXiv2017

Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshes

This work describes a domain embedding technique between two non-matching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is to apply the multilevel Monte Carlo (MLMC) method for the quantification of output uncertainties of PDEs with random input coefficients on general, unstructured computational domains. We propose a highly scalable, hierarchical sampling method to generate realizations of a Gaussian random field on a given unstructured mesh by solving a reaction-diffusion PDE with a stochastic right-hand side. The stochastic PDE is discretized using the mixed finite element method on an embedded domain with a structured mesh, and then the solution is projected onto the unstructured mesh. This work describes implementation details on how to efficiently transfer data from the structured and unstructured meshes at coarse levels, assuming this can be done efficiently on the finest level. We investigate the efficiency and parallel scalability of the technique for the scalable generation of Gaussian random fields in three dimensions. An application of the MLMC method is presented for quantifying uncertainties of subsurface flow problems. We demonstrate the scalability of the sampling method with non-matching mesh embedding, coupled with a parallel forward model problem solver, for large-scale 3D MLMC simulations with up to $1.9\cdot 10^9$ unknowns.

math.NA↗