Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application
No full textA thermodynamically consistent visco-elastodynamical model at finite strains is derived that also allows for inelasticity (like plasticity or creep), thermal coupling, and poroelasticity with diffusion. The theory is developed in the Eulerian framework and is shown to be consistent with the thermodynamic framework given by General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). For the latter we use that the transport terms are given in terms of Lie derivatives. Application is illustrated by two examples, namely volumetric phase transitions with dehydration in rocks and martensitic phase transitions in shape-memory alloys. A strategy towards a rigorous mathematical analysis is only very briefly outlined
Hybrid machine learning based scale bridging framework for permeability prediction of fibrous structures
Full text linkThis study introduces a hybrid machine learning-based scale-bridging framework for predicting the permeability of fibrous textile structures. By addressing the computational challenges inherent to multiscale modeling, the proposed approach evaluates the efficiency and accuracy of different scale-bridging methodologies combining traditional surrogate models and even integrating physics-informed neural networks (PINNs) with numerical solvers, enabling accurate permeability predictions across micro- and mesoscales. Four methodologies were evaluated: Single Scale Method (SSM), Simple Upscaling Method (SUM), Scale-Bridging Method (SBM), and Fully Resolved Model (FRM). SSM, the simplest method, neglects microscale permeability and exhibited permeability values deviating by up to 150\% of the FRM model, which was taken as ground truth at an equivalent lower fiber volume content. SUM improved predictions by considering uniform microscale permeability, yielding closer values under similar conditions, but still lacked structural variability. The SBM method, incorporating segment-based microscale permeability assignments, showed significant enhancements, achieving almost equivalent values while maintaining computational efficiency and modeling runtimes of ~45 minutes per simulation. In contrast, FRM, which provides the highest fidelity by fully resolving microscale and mesoscale geometries, required up to 270 times more computational time than SSM, with model files exceeding 300 GB. Additionally, a hybrid dual-scale solver incorporating PINNs has been developed and shows the potential to overcome generalization errors and the problem of data scarcity of the data-driven surrogate approaches. The hybrid framework advances permeability modelling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing
Parametric Modeling of Sparse Random Trees Using 3D Copulas
Full text linkWe provide a parametric modelling approach suitable for various kinds of hierarchical networks based on random geometric graphs. In these networks, we have two kinds of components, so-called high-level components (HLC) and low-level components (LLC). Each HLC is associated with a serving zone and all LLC within this area are connected to the corresponding HLC. So-called sparse LLC networks, where only few LLC occur in the typical serving zone, are a non-negligible subdomain when investigating hierarchical networks. Therefore, we supply distributional results for structural characteristics where two LLC are independently and uniformly distributed along the segment system of the typical serving zone. In particular, we are interested in the joint distribution of three quantities, namely the length of the joint part of the shortest paths from the LLC to the HLC as well as the lengths of the corresponding disjoint remaining parts. In order to provide a parametric, three-dimensional distribution function for these random variables, we utilise a pseudo-maximum likelihood approach. More precisely, we fit parametric approximation formulas to the marginal density functions as well as parametric copula functions which match with the observed correlation structure. We also provide an asymptotic result for the joint distribution of the connection lengths as the size of the typical cell increases unboundedly. This general modelling approach is explicitly explained for the case that the random geometric graph is formed by the edges of random tessellations
Chapter 14: Methods for Verifying Booked Capacities
No full textWe formalize the problem to verify the legal requirement that transport situations arising from booked capacity rights shall be technically feasible. In particular, we propose a stochastic version of the problem of verifying booked capacities together with two heuristic solution methods. These methods have been designed as decision-support tools for real-world usage by transmission system operators (TSOs). Our approach is based on combining a stochastic model with an adversarial model to an overall model for the transport situations requested by the transport customers. The first method is based on sampling to capture the stochastic information, whereas the second method uses multivariate quantiles for that purpose. Both methods generate a set of nominations that are checked for technical feasibility to arrive at an overall conclusion
Stressor-Induced Site Control of Quantum Dots for Single-Photon Sources
No full textThe strain field of selectively oxidized AlOx current apertures in an AlGaAs/GaAs mesa is utilized to define the nucleation site of InGaAs/GaAs quantum dots. A design is developed that allows for the self-aligned growth of single quantum dots in the center of a circular mesa. Measurements of the strain tensor applying transmission-electron holography yield excellent agreement with the calculated strain field. Single-dot spectroscopy of site-controlled dots proves narrow excitonic linewidth virtually free of spectral diffusion due to quantum-dot growth in a defect-free matrix. Implementation of such dots in an electrically driven pin structure yields single-dot electroluminescence. Single-photon emission with excellent purity is proved for this device using a Hanbury Brown and Twiss setup. The injection efficiency of the initial pin design is affected by a substantial lateral current spreading close to the oxide aperture. Applying 3D carrier-transport simulation a ppn doping profile is developed achieving a substantial improvement of the current injection
Solvability and optimal control of a multi-species Cahn--Hilliard--Keller--Segel tumor growth model
No full textThis paper investigates an optimal control problem associated with a two-dimensional multispecies Cahn–Hilliard–Keller–Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equations. The modeling derivation and corresponding analysis have been addressed in a previous contribution. Building on this foundation, the scope of this study involves investigating a distributed control problem with the goal of optimizing a tracking-type cost functional. This latter aims to minimize the deviation of tumor cell location from desired target configurations while penalizing the costs associated with implementing control measures, akin to introducing a suitable medication. Under appropriate mathematical assumptions, we demonstrate that sufficiently regular solutions exhibit continuous dependence on the control variable. Furthermore, we establish the existence of optimal controls and characterize the first-order necessary optimality conditions through a suitable variational inequality
Subdifferentials and penalty approximations of the obstacle problem
No full textWe consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalized problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving, for example, the positive part function max(0, \cdot ). Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem
hp-finite elements for elliptic optimal control problems with control constraints
No full textA distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. The adjoint control is associated with the residual of the variational inequality but does not appear in the weak formulation. Each of the three variables is discretized independently by hp-finite elements. In particular, the non-penetration condition of the control variable is relaxed to a finite set of quadrature points. Sufficient conditions for the unique existence of a discrete solution are stated. Also a priori error estimates and guaranteed convergence rates are derived in terms of the mesh size as well as of the polynomial degree. Moreover, reliable and efficient a posteriori error estimates are presented, which enable hp-adaptive mesh refinements. Several numerical experiments demonstrate the applicability of the discretization with hp-finite elements, the efficiency of the a posteriori error estimates and the improvements with respect to the convergence order resulting from the application of hp-adaptivity. In particular, the hp-adaptive schemes lead to superior convergence properties
Risk-averse optimal control of random elliptic variational inequalities
No full textWe consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem
A convex variational principle for the necessary conditions of classical optimal control
Full text linkA scheme for generating a family of convex variational principles is developed, the Euler--Lagrange equations of each member of the family formally corresponding to the necessary conditions of optimal control of a given system of ordinary differential equations (ODE) in a well-defined sense. The scheme is applied to the Quadratic-Quadratic Regulator problem for which an explicit form of the functional is derived, and existence of minimizers of the variational principle is rigorously shown. It is shown that the Linear-Quadratic Regulator problem with time-dependent forcing can be solved within the formalism without requiring any nonlinear considerations, in contrast to the use of a Riccati system in the classical methodology. Our work demonstrates a pathway for solving nonlinear control problems via convex optimization