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Numerical analysis of the SIMP model for the topology optimization problem of minimizing compliance in linear elasticity
No full textWe study the finite element approximation of the solid isotropic material with penalization method (SIMP) for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: W 1, p-type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to all the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding
A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application
Get PDFA 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
Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential
No full textThis paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn-Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the L1-norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before
An Eulerian approach to the regularized JKO scheme with low-rank tensor decompositions for Bayesian inversion
Get PDFThe possibility of using the Eulerian discretization for the problem of modelling high dimensional distributions and sampling, is studied. The problem is posed as a minimization problem over the space of probability measures with respect to the Wasserstein distance and solved with the entropy-regularized JKO scheme. Each proximal step can be formulated as a fixed-point equation and solved with accelerated methods, such as Anderson's. The usage of the low-rank Tensor Train format allows to overcome the curse of dimensionality, i.e. the exponential growth of degrees of freedom with dimension, inherent to Eulerian approaches. The resulting method requires only pointwise computations of the unnormalized posterior and is, in particular, gradient-free. Fixed Eulerian grid allows to employ a caching strategy, significally reducing the expensive evaluations of the posterior. When the Eulerian model of the target distribution is fitted, the passage back to the Lagrangian perspective can also be made, allowing to approximately sample from the distribution. We test our method both for synthetic target distributions and particular Bayesian inverse problems and report comparable or better performance than the baseline Metropolis-Hastings MCMC with the same amount of resources. Finally, the fitted model can be modified to facilitate the solution of certain associated problems, which we demonstrate by fitting an importance distribution for a particular quantity of interest. We release our code at https://github.com/viviaxenov/rJKOtt
Desynchronization of temporal solitons in Kerr cavities with pulsed injection.
Get PDFA numerical and analytical study was conducted to investigate the bifurcation mechanisms that cause desynchronization between the soliton repetition frequency and the frequency of external pulsed injection in a Kerr cavity described by the Lugiato-Lefever equation (LLE). The results suggest that desynchronization typically occurs through an Andronov-Hopf (AH) bifurcation. Additionally, a simple and intuitive criterion for this bifurcation to occur is proposed
Theory of the linewidth-power product of photonic-crystal surface-emitting lasers
Get PDFA general theory for the intrinsic (Lorentzian) linewidth of photonic--crystal surface--emitting lasers (PCSELs) is presented. The effect of spontaneous emission is modeled by a classical Langevin force entering the equation for the slowly varying waves. The solution of the coupled--wave equations, describing the propagation of four basic waves within the plane of the photonic crystal, is expanded in terms of the solutions of the associated spectral problem, i.e. the laser modes. Expressions are given for photon number, rate of spontaneous emission into the laser mode, Petermann factor and effective Henry factor entering the general formula for the linewidth. The theoretical framework is applied to the calculation of the linewidth--power product of air--hole and all--semiconductor PCSELs. For output powers in the Watt range, intrinsic linewidths in the kHz range are obtained in agreement with recent experimental results
Variational Structures Beyond Gradient Flows: a Macroscopic Fluctuation-Theory Perspective
No full textMacroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models
Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations
No full textConvection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation
Inf-Sup Stabilized Scott–Vogelius Pairs on General Shape-Regular Simplicial Grids for Navier–Stokes Equations
No full textThis paper considers the discretization of the time-dependent Navier–Stokes equations with the family of inf-sup stabilized Scott–Vogelius pairs recently introduced in [John/Li/Merdon/Rui, Math. Models Methods Appl. Sci., 2024] for the Stokes problem. Therein, the velocity space is obtained by enriching the -conforming Lagrange element space with some -conforming Raviart–Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a -like system for arbitrary velocity polynomial degree k. Numerical studies verify the theoretical findings
An Adaptive Stabilized Finite Element Method for the Stokes–Darcy Coupled Problem
No full textFor the Stokes–Darcy coupled problem, which models a fluid that flows from a free medium into a porous medium, we introduce and analyze an adaptive stabilized finite element method using Lagrange equal order element to approximate the velocity and pressure of the fluid. The interface conditions between the free medium and the porous medium are given by mass conservation, the balance of normal forces, and the Beavers–Joseph–Saffman conditions. We prove the well-posedness of the discrete problem and present a convergence analysis with optimal error estimates in natural norms. Next, we introduce and analyze a residual-based a posteriori error estimator for the stabilized scheme. Finally, we present numerical examples to demonstrate the performance and effectiveness of our scheme