Patrick Dondl
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View article: Numerical Approximation of the logarithmic Laplacian via sinc-basis
Numerical Approximation of the logarithmic Laplacian via sinc-basis Open
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 fractiona…
View article: Arbitrary precision computation of hydrodynamic stability eigenvalues
Arbitrary precision computation of hydrodynamic stability eigenvalues Open
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 hydrodyn…
View article: Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs
Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs Open
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 pa…
View article: Thermal Convection in a Linearly Viscous Fluid Overlying a Bidisperse Porous Medium
Thermal Convection in a Linearly Viscous Fluid Overlying a Bidisperse Porous Medium Open
A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macropores but also cracks or fissures in the skeleton which give rise to micropores. In this article, we develop and analyze a model for therma…
View article: Non-local homogenization limits of discrete elastic spring network models with random coefficients
Non-local homogenization limits of discrete elastic spring network models with random coefficients Open
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 behavi…
View article: Phase Field Model for Multi-Material Shape Optimization of Inextensible Rods
Phase Field Model for Multi-Material Shape Optimization of Inextensible Rods Open
We derive a model for the optimization of the bending and torsional rigidities of nonhomogeneous elastic rods. This is achieved by studying a sharp interface shape optimization problem with perimeter penalization, that treats both rigiditi…
View article: Γ-convergence of a discrete Kirchhoff rod energy
Γ-convergence of a discrete Kirchhoff rod energy Open
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 Γ-convergence to the continuous model. This…
View article: $Γ$-convergence of a discrete Kirchhoff rod energy
$Γ$-convergence of a discrete Kirchhoff rod energy Open
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 $Γ$-convergence to the continuous model. Thi…
View article: Efficient uncertainty quantification for mechanical properties of randomly perturbed elastic rods
Efficient uncertainty quantification for mechanical properties of randomly perturbed elastic rods Open
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 elas…
View article: Analysis of a sinc-Galerkin Method for the Fractional Laplacian
Analysis of a sinc-Galerkin Method for the Fractional Laplacian Open
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 me…
View article: Phase field model for multi-material shape optimization of inextensible rods
Phase field model for multi-material shape optimization of inextensible rods Open
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 rigidit…
View article: Eya-controlled affinity between cell lineages drives tissue self-organization during <i>Drosophila</i> oogenesis
Eya-controlled affinity between cell lineages drives tissue self-organization during <i>Drosophila</i> oogenesis Open
The cooperative morphogenesis of cell lineages underlies the development of functional units and organs. To study mechanisms driving the coordination of lineages, we investigated soma-germline interactions during oogenesis. From invertebra…
View article: Viscosity solutions for doubly nonlinear evolution equations
Viscosity solutions for doubly nonlinear evolution equations Open
We extend the theory of viscosity solutions to treat scalar-valued doubly-nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontin…
View article: Charting the twist-to-bend ratio of plant axes
Charting the twist-to-bend ratio of plant axes Open
During the evolution of land plants many body plans have been developed. Differences in the cross-sectional geometry and tissue pattern of plant axes influence their flexural rigidity, torsional rigidity and the ratio of both of these rigi…
View article: Variational modeling of paperboard delamination under bending
Variational modeling of paperboard delamination under bending Open
We develop and analyze a variational model for laminated 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 constr…
View article: Data from Charting the twist-to-bend ratio of plant axes
Data from Charting the twist-to-bend ratio of plant axes Open
During evolution of land plants many body plans have been developed. Differences in the cross-sectional geometry and tissue pattern of plant axes influence their flexural rigidity, torsional rigidity and the ratio of these both rigidities,…
View article: Linearization and computation for large-strain visco-elasticity
Linearization and computation for large-strain visco-elasticity Open
Time-discrete numerical minimization schemes for simple visco-elastic materials in the Kelvin-Voigt rheology at high strains are not well posed because of the non-quasi-convexity of the dissipation functional. A possible solution is to res…
View article: Three Dimensional Optimization of Scaffold Porosities for Bone Tissue Engineering
Three Dimensional Optimization of Scaffold Porosities for Bone Tissue Engineering Open
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 pr…
View article: $L^p(I,C^α(Ω))$ Regularity for Reaction-Diffusion Equations with Non-smooth Data
$L^p(I,C^α(Ω))$ Regularity for Reaction-Diffusion Equations with Non-smooth Data Open
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 …
View article: Surface lattice Green’s functions for high-entropy alloys
Surface lattice Green’s functions for high-entropy alloys Open
We study the surface elastic response of pure Ni, the random alloy FeNiCr and an average FeNiCr alloy in terms of the surface lattice Green’s function. We propose a scheme for computing per-site Green’s function and study their per-site va…
View article: Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear Problems
Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear Problems Open
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 boundar…
View article: Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear\n Problems
Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear\n Problems Open
We provide convergence guarantees for the Deep Ritz Method for abstract\nvariational energies. Our results cover non-linear variational problems such as\nthe $p$-Laplace equation or the Modica-Mortola energy with essential or natural\nboun…
View article: Linearization and Computation for Large-Strain Viscoelasticity
Linearization and Computation for Large-Strain Viscoelasticity Open
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-…
View article: Thermal convection in a linearly viscous fluid overlying a bidisperse porous medium
Thermal convection in a linearly viscous fluid overlying a bidisperse porous medium Open
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 therm…
View article: Variational Modeling of Paperboard Delamination Under Bending
Variational Modeling of Paperboard Delamination Under Bending Open
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 prov…