arXiv subjects

Barbara Wohlmuth

Publications and source records attributed to Barbara Wohlmuth.

At least 73 records · Page 4Linked to original sources

arXiv2020

A 3D-1D coupled blood flow and oxygen transport model to generate microvascular networks

In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine-scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D-1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi-directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects.

cs.CE↗

arXiv2020

The surrogate matrix methodology: Accelerating isogeometric analysis of waves

The surrogate matrix methodology delivers low-cost approximations of matrices (i.e., surrogate matrices) which are normally computed in Galerkin methods via element-scale quadrature formulas. In this paper, the methodology is applied to a number of model problems in wave mechanics treated in the Galerkin isogeometic setting. Herein, the resulting surrogate methods are shown to significantly reduce the assembly time in high frequency wave propagation problems. In particular, the assembly time is reduced with negligible loss in solution accuracy. This paper also extends the scope of previous articles in its series by considering multi-patch discretizations of time-harmonic, transient, and nonlinear PDEs as particular use cases of the methodology. Our a priori error analysis for the Helmholtz equation demonstrates that the additional consistency error introduced by the presence of surrogate matrices is independent of the wave number. In addition, our floating point analysis establishes that the computational complexity of the methodology compares favorably to other contemporary fast assembly techniques for isogeometric methods. Our numerical experiments demonstrate clear performance gains for time-harmonic problems, both with and without the presence of perfectly matched layers. Notable speed-ups are also presented for a transient problem with a compressible neo-Hookean material.

math.NA↗

arXiv2020

Stencil scaling for vector-valued PDEs on hybrid grids with applications to generalized Newtonian fluids

Matrix-free finite element implementations for large applications provide an attractive alternative to standard sparse matrix data formats due to the significantly reduced memory consumption. Here, we show that they are also competitive with respect to the run time in the low order case if combined with suitable stencil scaling techniques. We focus on variable coefficient vector-valued partial differential equations as they arise in many physical applications. The presented method is based on scaling constant reference stencils originating from a linear finite element discretization instead of evaluating the bilinear forms on-the-fly. This method assumes the usage of hierarchical hybrid grids, and it may be applied to vector-valued second-order elliptic partial differential equations directly or as a part of more complicated problems. We provide theoretical and experimental performance estimates showing the advantages of this new approach compared to the traditional on-the-fly integration and stored matrix approaches. In our numerical experiments, we consider two specific mathematical models. Namely, linear elastostatics and incompressible Stokes flow. The final example considers a non-linear shear-thinning generalized Newtonian fluid. For this type of non-linearity, we present an efficient approach to compute a regularized strain rate which is then used to define the node-wise viscosity. Depending on the compute architecture, we could observe maximum speedups of 64% and 122% compared to the on-the-fly integration. The largest considered example involved solving a Stokes problem with 12288 compute cores on the state of the art supercomputer SuperMUC-NG.

cs.CE↗

arXiv2020

Generalized bounds for active subspaces

In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincaré inequalities are not valid due to unbounded Poincaré constants. Consequently, we propose a framework that allows to derive generalized estimates in the sense that it enables to control the trade-off between the size of the Poincaré constant and a weaker order of the final error bound. In particular, we investigate independently exponentially distributed random variables in dimension two or larger and give explicit expressions for corresponding Poincaré constants showing their dependence on the dimension of the problem. Finally, we suggest possibilities for future work that aim for extending the class of distributions applicable to the active subspace method as we regard this as an opportunity to enlarge its usability.

math.PR↗

arXiv2019

A high-order discontinuous Galerkin method for nonlinear sound waves

We propose a high-order discontinuous Galerkin scheme for nonlinear acoustic waves on polytopic meshes. To model sound propagation with and without losses, we use Westervelt's nonlinear wave equation with and without strong damping. Challenges in the numerical analysis lie in handling the nonlinearity in the model, which involves the derivatives in time of the acoustic velocity potential, and in preventing the equation from degenerating. We rely in our approach on the Banach fixed-point theorem combined with a stability and convergence analysis of a linear wave equation with a variable coefficient in front of the second time derivative. By doing so, we derive an a priori error estimate for Westervelt's equation in a suitable energy norm for the polynomial degree $p \geq 2$. Numerical experiments carried out in two-dimensional settings illustrate the theoretical convergence results. In addition, we demonstrate efficiency of the method in a three-dimensional domain with varying medium parameters, where we use the discontinuous Galerkin approach in a hybrid way.

math.NA↗

arXiv2019

The surrogate matrix methodology: Low-cost assembly for isogeometric analysis

A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computational domain. This, in turn, determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. Poisson's equation, membrane vibration, plate bending, and Stokes' flow problems are studied. In these problems, the use of surrogate matrices has a negligible impact on solution accuracy. Because only a small fraction of the original quadrature must be performed, we are able to report beyond a fifty-fold reduction in overall assembly time in the same software. The capacity for even further speed-ups is clearly demonstrated. The implementation used here was achieved by a small number of modifications to the open-source IGA software library GeoPDEs. Similar modifications could be made to other present-day software libraries.

math.NA↗

arXiv2019

The surrogate matrix methodology: A reference implementation for low-cost assembly in isogeometric analysis

A reference implementation of a new method in isogeometric analysis (IGA) is presented. It delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed by element-scale quadrature. To generate surrogate matrices, quadrature must only be performed on a fraction of the elements in the computational domain. In this way, quadrature determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. We present the modifications and extensions required for a reference implementation in the open-source IGA software library GeoPDEs. The exposition is fashioned to help facilitate similar modifications in other contemporary software libraries.

cs.MS↗

arXiv2019

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law while slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complimentary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2d and 3d, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.

math.NA↗

arXiv2019

Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces

In this article, we perform a parameter study for a recently developed karst hydrological model. The study consists of a high-dimensional Bayesian inverse problem and a global sensitivity analysis. For the first time in karst hydrology, we use the active subspace method to find directions in the parameter space that dominate the Bayesian update from the prior to the posterior distribution in order to effectively reduce the dimension of the problem and for computational efficiency. Additionally, the calculated active subspace can be exploited to construct sensitivity metrics on each of the individual parameters and be used to construct a natural model surrogate. The model consists of 21 parameters to reproduce the hydrological behavior of spring discharge in a karst aquifer located in the Kerschbaum spring recharge area at Waidhofen a.d. Ybbs in Austria. The experimental spatial and time series data for the inference process were collected by the water works in Waidhofen. We show that this case study has implicit low-dimensionality, and we run an adjusted Markov chain Monte Carlo algorithm in a low-dimensional subspace to construct samples of the posterior distribution. The results are visualized and verified by plots of the posterior's push-forward distribution displaying the uncertainty in predicting discharge values due to the experimental noise in the data. Finally, a discussion provides hydrological interpretation of these results for the Kerschbaum area.

stat.CO↗

arXiv2019

A statistical framework for generating microstructures of two-phase random materials: application to fatigue analysis

Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigue analysis usually employ simple microstructure, consisting of ideal geometries such as ellipsoidal inclusions, which generally does not capture complex geometries. In this paper, we introduce a simple but flexible surrogate microstructure model for two-phase materials through a level-cut of a Gaussian random field with covariance of Matérn class. Such parametrization of the covariance function allows for the representation of a few key design parameters while representing the geometry of inclusions in a more general setting for a large class of random heterogeneous two-phase media. In addition to the traditional morphology descriptors such as porosity, size and aspect ratio, it provides control of the regularity of the inclusions interface and sphericity. These parameters are estimated from a small number of real material images using Bayesian inversion. An efficient process of evaluating the samples, based on the Fast Fourier Transform, makes possible the use of Monte-Carlo methods to estimate statistical properties for the quantities of interest in a given material class. We demonstrate the overall framework of the use of the surrogate material model in application to the uncertainty quantification in fatigue analysis, its feasibility and efficiency, and its role in the microstructure design.

physics.app-ph↗

arXiv2019

Hybrid models for simulating blood flow in microvascular networks

In this paper, we are concerned with the simulation of blood flow in microvascular networks and the surrounding tissue. To reduce the computational complexity of this issue, the network structures are modeled by a one-dimensional graph, whose location in space is determined by the centerlines of the three-dimensional vessels. The surrounding tissue is considered as a homogeneous porous medium. Darcy's equation is used to simulate flow in the extra-vascular space, where the mass exchange with the blood vessels is accounted for by means of line source terms. However, this model reduction approach still causes high computational costs, in particular, when larger parts of an organ have to be simulated. This observation motivates the consideration of a further model reduction step. Thereby, we homogenize the fine scale structures of the microvascular networks resulting in a new hybrid approach modeling the fine scale structures as a heterogeneous porous medium and the flow in the larger vessels by one-dimensional flow equations. Both modeling approaches are compared with respect to mass fluxes and averaged pressures. The simulations have been performed on a microvascular network that has been extracted from the cortex of a rat brain.

math.NA↗

arXiv2019

On the unsteady Darcy-Forchheimer-Brinkman equation in local and nonlocal tumor growth models

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases.

math.AP↗

arXiv2019

Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation

We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of MDE and ECM, together with chemotaxis, haptotaxis, apoptosis, nutrient distribution, and cell-to-matrix adhesion. We provide a rigorous proof of the existence of solutions of the coupled system with gradient-based and adhesion-based haptotaxis effects. In addition, we discuss finite element discretizations of the model, and we present the results of numerical experiments designed to show the relative importance and roles of various effects, including cell mobility, proliferation, necrosis, hypoxia, and nutrient concentration on the generation of MDEs and the degradation of the ECM.

math.AP↗

arXiv2019

An all-at-once Newton strategy for methane hydrate reservoir models

Marine gas hydrate systems are characterized by highly dynamic transport-reaction processes in an essentially water-saturated porous medium that are coupled to thermodynamic phase transitions between solid gas hydrates, free gas and dissolved methane in the aqueous phase. These phase transitions are highly nonlinear and strongly coupled, and cause the mathematical model to rapidly switch the phase states and pose serious convergence issues for the classical Newton's method. One of the common methods of dealing with such phase transitions is the primary variable switching (PVS) method where the choice of the primary variables is adapted locally to the phase state **outside** the Newton loop. In order to ensure that the phase states are determined accurately, the PVS strategy requires an additional iterative loop, which can get quite expensive for highly nonlinear problems. For methane hydrate reservoir models, the PVS method shows poor convergence behaviour and often leads to extremely small time step sizes. In order to overcome this issue, we have developed a nonlinear complementary constraints method (NCP) for handling phase transitions, and implemented it within a non-smooth Newton's linearization scheme using an active-set strategy. Here, we present our numerical scheme and show its robustness through field scale applications based on the highly dynamic geological setting of the Black Sea.

physics.comp-ph↗

arXiv2019

A priori error estimates for the finite element approximation of Westervelt's quasilinear acoustic wave equation

We study the spatial discretization of Westervelt's quasilinear strongly damped wave equation by piecewise linear finite elements. Our approach employs the Banach fixed-point theorem combined with a priori analysis of a linear wave model with variable coefficients. Degeneracy of the semi-discrete Westervelt equation is avoided by relying on the inverse estimates for finite element functions and the stability and approximation properties of the interpolation operator. In this way, we obtain optimal convergence rates in $L^2$-based spatial norms for sufficiently small data and mesh size and an appropriate choice of initial approximations. Numerical experiments in a setting of a 1D channel as well as for a focused-ultrasound problem illustrate our theoretical findings.

math.NA↗

arXiv2019

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

math.NA↗

arXiv2019

Self-adaptive absorbing boundary conditions for quasilinear acoustic wave propagation

We propose a self-adaptive absorbing technique for quasilinear ultrasound waves in two- and three-dimensional computational domains. As a model for the nonlinear ultrasound propagation in thermoviscous fluids, we employ Westervelt's wave equation solved for the acoustic velocity potential. The angle of incidence of the wave is computed based on the information provided by the wave-field gradient which is readily available in the finite element framework. The absorbing boundary conditions are then updated with the angle values in real time. Numerical experiments illustrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2019

Goal-Oriented Adaptive Modeling of Random Heterogeneous Media and Model-Based Multilevel Monte Carlo Methods

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.

math.NA↗