arXiv subjects

Patrick Dondl

Publications and source records attributed to Patrick Dondl.

At least 19 recordsLinked to original sources

arXiv2025

Numerical Approximation of the logarithmic Laplacian via sinc-basis

In recent works, the authors of this chapter have shown with co-authors how a basis consisting of dilated and shifted $\text{sinc}$-functions can be used to solve fractional partial differential equations. As a model problem, the fractional Dirichlet problem with homogeneous exterior value conditions was solved. In this work, we briefly recap the algorithms developed there and that -- from a computational point of view -- they can be used to solve nonlocal equations given through different operators as well. As an example, we numerically solve the Dirichlet problem for the logarithmic Laplacian $\log(-\Delta)$ which has the Fourier symbol $\log(\left|\omega\right|^2)$ and compute its Eigenvalues on disks with different radii in $\mathbb R^2$.

math.NA↗

arXiv2025

Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces

The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow.

math.NA↗

arXiv2025

Arbitrary precision computation of hydrodynamic stability eigenvalues

We show that by using higher order precision arithmetic, i.e., using floating point types with more significant bits than standard double precision numbers, one may accurately compute eigenvalues for non-normal matrices arising in hydrodynamic stability problems. The basic principle is illustrated by a classical example of two $7\times 7$ matrices for which it is well known that eigenvalue computations fail when using standard double precision arithmetic. We then present an implementation of the Chebyshev tau-QZ method allowing the use of a large number of Chebyshev polynomials together with arbitrary precision arithmetic. This is used to compute the behavior of the spectra for Couette and Poiseuille flow at high Reynolds number. An experimental convergence analysis finally makes it evident that high order precision is required to obtain accurate results.

math.NA↗

arXiv2025

Micro-Macro Coupling for Optimizing Scaffold Mediated Bone Regeneration

This work presents a framework for modeling three-dimensional scaffold-mediated bone regeneration and the associated optimization problem. By incorporating microstructure into the model through periodic homogenization, we capture the effects of microscale fluctuations on the bone growth process. Numerical results and optimized scaffold designs that explicitly account for the microstructure are presented, demonstrating the potential of this approach for improving scaffold performance.

math.NA↗

arXiv2024

Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations. We obtain the singular limit of the evolution equations as the length parameter of the phase fields tends to zero by formal expansions and numerically confirm its validity for circles in two dimensions. Our analysis is complemented by numerical experiments for planar curves, surfaces in three-dimensional space, and semi-supervised learning tasks on graphs.

math.AP↗

arXiv2024

Non-local homogenization limits of discrete elastic spring network models with random coefficients

This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers.

math.AP↗

arXiv2023

$\Gamma$-convergence of a discrete Kirchhoff rod energy

This work is motivated by the classical discrete elastic rod model by Audoly et al. We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove $\Gamma$-convergence to the continuous model. This discrete energy is given by the bending and torsion energy of an interpolating conforming polynomial curve and provides a simple formula for the bending energy depending in each discrete segment only on angle and adjacent edge lengths. For the $\liminf$-inequality, we need to introduce penalty terms to ensure arc-length parametrization in the limit. For the recovery sequence a discretization with equal Euclidean distance between consecutive points is constructed. Particular care is taken to treat the interaction between bending and torsion by employing a discrete version of the Bishop frame.

math.AP↗

arXiv2023

Efficient uncertainty quantification for mechanical properties of randomly perturbed elastic rods

Motivated by an application involving additively manufactured bioresorbable polymer scaffolds supporting bone tissue regeneration, we investigate the impact of uncertain geometry perturbations on the effective mechanical properties of elastic rods. To be more precise, we consider elastic rods modeled as three-dimensional linearly elastic bodies occupying randomly perturbed domains. Our focus is on a model where the cross-section of the rod is shifted along the longitudinal axis with stationary increments. To efficiently obtain accurate estimates on the resulting uncertainty of the effective elastic moduli, we use a combination of analytical and numerical methods. Specifically, we rigorously derive a one-dimensional surrogate model by analyzing the slender-rod $\Gamma$-limit. Additionally, we establish qualitative and quantitative stochastic homogenization results for the one-dimensional surrogate model. To compare the fluctuations of the surrogate with the original three-dimensional model, we perform numerical simulations by means of finite element analysis and Monte Carlo methods.

math.AP↗

arXiv2022

Analysis of a sinc-Galerkin Method for the Fractional Laplacian

We provide the convergence analysis for a sinc-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of an earlier article by the same authors, where the authors presented a sinc-function based method to solve fractional PDEs. While the original method was formulated as a collocation method, we show that the same method can be interpreted as a nonconforming Galerkin method, giving access to abstract error estimates. Optimal order of convergence is shown without any unrealistic regularity assumptions on the solution.

math.NA↗

arXiv2022

Phase field model for multi-material shape optimization of inextensible rods

We derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigidities as objectives. We then formulate a phase field approximation of the optimization problem and show the convergence to the aforementioned sharp interface model via $\Gamma$-convergence. In the final part of this work we numerically approximate minimizers of the phase field problem by using a steepest descent approach and relate the resulting optimal shapes to the development of the morphology of plant stems.

math.OC↗

arXiv2021

Three Dimensional Optimization of Scaffold Porosities for Bone Tissue Engineering

We consider the scaffold design optimization problem associated to the three dimensional, time dependent model for scaffold mediated bone regeneration considered in Dondl et al. (2021). We prove existence of optimal scaffold designs and present numerical evidence that optimized scaffolds mitigate stress shielding effects from exterior fixation of the scaffold at the defect site.

math.AP↗

arXiv2021

$L^p(I,C^α(Ω))$ Regularity for Reaction-Diffusion Equations with Non-smooth Data

We prove an $L^p(I,C^α(Ω))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^α(Ω))$ norm with respect to the data. To prove our result, we first establish $C^α(Ω)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).

math.AP↗

arXiv2021

Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear Problems

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover non-linear variational problems such as the $p$-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across % bounded families of right-hand sides.

math.NA↗

arXiv2021

Linearization and Computation for Large-Strain Viscoelasticity

Time-discrete numerical minimization schemes for simple viscoelastic materials in the large strain Kelvin-Voigt rheology are not well-posed due to non-quasiconvexity of the dissipation functional. A possible solution is to resort into non-simple material models with higher-order gradients of deformations. This makes, however, numerical computations much more involved. Here we propose another approach relying on local minimizers of the simple-material model. Computational tests are provided showing a very good agreement between our model and the original one.

math.NA↗

arXiv2021

Thermal convection in a linearly viscous fluid overlying a bidisperse porous medium

A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macro pores but also cracks or fissures in the skeleton which give rise to micro pores. In this article we develop and analyse a model for thermal convection where a layer of viscous incompressible fluid overlies a layer of bidisperse porous medium. Care has to be taken with the boundary conditions at the interface of the fluid and the porous material and this aspect is investigated. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are found. These depend on the ratio ${\hat d}$ of the depth $d$ of the fluid layer to the depth $d_m$ of the porous layer. We show that there is a critical value of ${\hat d}$ such that below this value convective motion initiates in the porous layer whereas for ${\hat d}$ above this value the convective instability commences in the fluid layer.

physics.flu-dyn↗

arXiv2021

Variational Modeling of Paperboard Delamination Under Bending

We develop and analyze a variational model for multi-ply (i.e., multi-layered) paperboard. The model consists of a number of elastic sheets of a given thickness, which -- at the expense of an energy per unit area -- may delaminate. By providing an explicit construction for possible admissible deformations subject to boundary conditions that introduce a single bend, we discover a rich variety of energetic regimes. The regimes correspond to the experimentally observed: initial purely elastic response for small bending angle and the formation of a localized inelastic, delaminated hinge once the angle reaches a critical value. Our scaling upper bound then suggests the occurrence of several additional regimes as the angle increases. The upper bounds for the energy are partially matched by scaling lower bounds.

math.AP↗

arXiv2021

A parameter study on optimal scaffolds in a simple model for bone regeneration

We propose a simple model for scaffold aided bone regeneration. In this model, only macroscopic quantities, e.g., locally averaged osteoblast densities, are considered. This allows for use of this model in an optimization algorithm, whose outcome is an optimal scaffold porosity distribution. This optimal scaffold naturally depends on the choice of parameters in the model, and we provide a parameter study with a particular focus on patients with reduced bone regeneration or reduced vascularization capacity.

q-bio.TO↗

arXiv2021

Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis

Fueled by many applications in random processes, imaging science, geophysics, etc., fractional Laplacians have recently received significant attention. The key driving force behind the success of this operator is its ability to capture non-local effects while enforcing less smoothness on functions. In this paper, we introduce a spectral method to approximate this operator employing a sinc basis. Using our scheme, the evaluation of the operator and its application onto a vector has complexity of $\mathcal O(N\log(N))$ where $N$ is the number of unknowns. Thus, using iterative methods such as CG, we provide an efficient strategy to solve fractional partial differential equations with exterior Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works in both $2d$ and $3d$. We also recover the FEM rates of convergence on benchmark problems. We further illustrate the efficiency of our approach by applying it to fractional Allen-Cahn and image denoising problems.

math.NA↗