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Reduced basis methods for efficient simulation of a rigid robot hand interacting with soft tissue

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arXiv 2021年

作者： Shuva, Shahnewaz Buchfink, Patrick Röhrle, Oliver Haasdonk, Bernard Institute of Applied Analysis and Numerical Simulation University of Stuttgart Germany Institute for Modeling and Simulation of Biomechanical Systems University of Stuttgart Germany

We present efficient reduced basis (RB) methods for the simulation of a coupled problem consisting of a rigid robot hand interacting with soft tissue material. The soft tissue is modeled by the linear elasticity equation and discretized with the Finite Element Method. We look at two different scenarios: (i) the forward simulation and (ii) a feedback control formulation of the model. In both cases, large-scale systems of equations appear, which need to be solved in real-time. This is essential in practice for the implementation in a real robot. For the feedback-scenario, we encounter a high-dimensional Algebraic Riccati Equation (ARE) in the context of the linear quadratic regulator. To overcome the real-time constraint by significantly reducing the computational complexity, we use several structure-preserving and non-structure-preserving reduction methods. These include reduced basis techniques based on the Proper Orthogonal Decomposition. For the ARE, we compute a low-rank-factor and hence solve a low-dimensional ARE instead of solving a full dimensional problem. numerical examples for both cases (i) and (ii) are provided. These illustrate the approximation quality of the reduced solution and speedup factors of the different reduction approaches. Copyright © 2021, The Authors. All rights reserved.

关键词： Real time systems

Residual-Based a Posteriori Error Estimators for Algebraic Stabilizations

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arXiv 2024年

作者： Jha, Abhinav Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes [6] to the newly proposed algebraic stabilization schemes [9, 10]. numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme. © 2024, CC BY.

关键词： Finite element method

A GENERALIZED THREE LINES LEMMA IN HARDY-LIKE SPACES

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arXiv 2024年

作者： Corso, Thiago Carvalho Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

In this paper we address the following question: given a holomorphic function with prescribed Lp() and Lq() norm (with 1 ≤ p,q ≤ ∞) along two parallel lines in the complex plane, then what is the maximum value that this function can achieve at a given point between these lines. Here we show that this problem is well-posed in suitable Hardy-like spaces on the strip. Moreover, in this setting we completely solve this problem by providing not only an explicit formula for the optimizers but also for the optimal values. In addition, we briefly discuss some applications of these results to interpolation theory and to Lieb-Thirring *** Codes Primary: 30C70 Secondary: 30H10, 30C80, 35P15, 81Q10 © 2024, CC BY-NC-SA.

关键词： Eigenvalues and eigenfunctions

Dictionary-based Online-adaptive Structure-preserving Model Order Reduction for Parametric Hamiltonian Systems

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arXiv 2023年

作者： Herkert, Robin Buchfink, Patrick Haasdonk, Bernard Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e. the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach. Copyright © 2023, The Authors. All rights reserved.

关键词： Computational complexity

On darcy-and brinkman-type models for two-phase flow in asymptotically flat domains

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arXiv 2017年

作者： Armiti-Juber, Alaa Rohde, Christian Institute for Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in [Guo et al., 2014] or the asymptotic approach in [Yortsos, 1995]. The latter approach uses a geometrical scaling. We show the efficiency of the approach in asymptotically flat domains for Darcy regimes. Moreover, we prove that it is vastly equivalent to the multiscale model approach. We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws [Hassanizadeh and Gray, 1993]. Copyright © 2017, The Authors. All rights reserved.

关键词： Porous materials

Stokes–Brinkman–Darcy models for fluid–porous systems: derivation, analysis and validation

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arXiv 2024年

作者： Ruan, Linheng Rybak, Iryna Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Flow interaction between a plain-fluid region in contact with a porous layer attracted significant attention from modelling and analysis sides due to numerous applications in biology, environment and industry. In the most widely used coupled model, fluid flow is described by the Stokes equations in the free-flow domain and Darcy’s law in the porous medium, and complemented by the appropriate interface conditions. However, traditional coupling concepts are restricted, with a few exceptions, to one-dimensional flows parallel to the fluid–porous interface. In this work, we use an alternative approach to model interaction between the plain-fluid domain and porous medium by considering a transition zone, and propose the full- and hybrid-dimensional Stokes–Brinkman–Darcy models. In the first case, the equi-dimensional Brinkman equations are considered in the transition region, and the appropriate interface conditions are set on the top and bottom of the transition zone. In the latter case, we perform a dimensional model reduction by averaging the Brinkman equations in the normal direction and using the proposed transmission conditions. The well-posedness of both coupled problems is proved, and some numerical simulations are carried out in order to validate the *** Codes 35Q35, 65N08, 76B03, 76D07, 76S05 Copyright © 2024, The Authors. All rights reserved.

关键词： Porous materials

analysis of the Stokes-Darcy problem with generalised interface conditions

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arXiv 2021年

作者： Eggenweiler, Elissa Discacciati, Marco Rybak, Iryna University of Stuttgart Institute of Applied Analysis and Numerical Simulation Loughborough University Department of Mathematical Sciences

Fluid flows in coupled systems consisting of a free-flow region and the adjacent porous medium appear in a variety of environmental settings and industrial applications. In many applications, fluid flow is non-parallel to the fluid-porous interface that requires a generalisation of the Beavers-Joseph coupling condition typically used for the Stokes-Darcy problem. Generalised coupling conditions valid for arbitrary flow directions to the interface are recently derived using the theory of homogenisation and boundary layers. The aim of this work is the mathematical analysis of the Stokes-Darcy problem with these generalised interface conditions. We prove the existence and uniqueness of the weak solution of the coupled problem. The well-posedness is guaranteed under a suitable relationship between the permeability and the boundary layer constants containing geometrical information about the porous medium and the interface. We numerically study the validity of the obtained results for realistic problems and provide a benchmark for numerical solution of the Stokes-Darcy problem with generalised interface *** Codes 35Q35, 65N08, 76D03, 76D07, 76S05 Copyright © 2021, The Authors. All rights reserved.

关键词： Boundary layers

Effective coupling conditions for arbitrary flows in Stokes–Darcy systems

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arXiv 2020年

作者： Eggenweiler, Elissa Rybak, Iryna University of Stuttgart Institute of Applied Analysis and Numerical Simulation Pfaffenwaldring 57 Stuttgart70569 Germany

Boundary conditions at the interface between the free-flow region and the adjacent porous medium is a key issue for physically consistent modeling and accurate numerical simulation of flow and transport processes in coupled systems due to the interface driven nature of such processes. Interface conditions available in the literature have several weak points: most of them are suitable only for flows parallel to the fluid–porous interface, some are restricted to specific boundary value problems, and others contain unknown model parameters which still need to be deter-mined. These facts severely restrict the variety of applications that can be successfully modeled. We propose new interface conditions which are valid for arbitrary flow directions. These coupling conditions are rigorously derived using the theory of homogenization and boundary layers. All effective parameters appearing in these conditions are computed numeri-cally based on the geometrical configuration of the coupled system. The developed conditions are validated by comparison of numerical simulation results for the coupled Stokes–Darcy model and the pore-scale resolved model. In addition, the new interface conditions are compared with the classical conditions to demonstrate the advantage of the proposed *** Codes 35Q35, 76D07, 76M10, 76M50, 76S05 Copyright © 2020, The Authors. All rights reserved.

关键词： Porous materials

REDUCED-DIMENSIONAL MODELLING FOR NONLINEAR CONVECTION-DOMINATED FLOW IN CYLINDRIC DOMAINS

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arXiv 2024年

作者： Mel’Nyk, Taras Rohde, Christian Institute of Applied Analysis and Numerical Simulation The University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection-diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear Robin-type boundary condition involving convective and diffusive fluxes is imposed on the lateral surface. The limit problem for vanishing diffusion and the cylinder shrinking to an interval is a nonlinear first-order conservation law. For a time span that allows for a classical solution of this limit problem corresponding uniform pointwise and energy estimates are proven. They provide precise model error estimates with respect to the small parameter that controls the double viscosity-geometric limit. In addition, other problems with more higher Péclet numbers are also *** Codes 76R50, 35K59, 35B40, 35B25 Copyright © 2024, The Authors. All rights reserved.

关键词： Nonlinear equations