Workshop with the Kanazawa University group 2024

Time and Venue

Schedule

• 09:00 - 09:30 Hirofumi Notsu (Kanazawa University): An energy estimate and a stabilized Lagrange-Galerkin scheme for a multiphase flow model
  Abstract: Multiphase flow models are commonly employed for understanding complex fluid flows. In this talk, we prove an energy decay property for a multiphase flow model in [Gidaspow, Multiphase Flow and Fluidization, 1994] and propose a stabilized Lagrange– Galerkin scheme. Here, a hyperbolic tangent transformation is employed to preserve the boundedness of the volume fraction, and a novel artificial term is added to obtain the stability property. Two-dimensional numerical examples are presented to see the experimental order of convergence and applicability in modeling sedimentation phenomena. This talk is based on the paper [A. Rudiawan, A. Zak, M. Benes, M. Kimura, and H. Notsu. An energy estimate and a stabilized Lagrange-Galerkin scheme for a multiphase flow model. Applied Mathematics Letters, 153:109059, 2024].
• 09:30 - 10:00 Masato Kimura (Kanazawa University): Well-posedness of a moving boundary problem related to an inverse shape determination problem
  Abstract: The purpose of this study is twofold. First, we revisit a shape optimization reformulation of a shape inverse problem and propose a simple and efficient numerical approach for realizing the minimization problem. Second, we examine the short-time existence and uniqueness of a classical solution to a Hele-Shaw-like system derived from the shape optimization formulation of the shape inverse problem. This is a joint work with Dr. Julius Fergy T. Rabago, Kanazawa University.
• 10:00 - 10:30 Coffee break
• 10:30 - 11:00 Koya Sakakibara (Kanazawa University): Advanced numerical analysis of the Plateau problem using the method of fundamental solutions
  Abstract: This presentation introduces a novel high-speed, high-accuracy computational approach for numerically solving the Plateau problem to find minimal surfaces bounded by a given contour. The proposed method employs the Method of Fundamental Solutions (MFS), a mesh-free approach that constructs smooth approximations to minimal surfaces without requiring traditional meshing techniques. Unlike conventional methods, our MFS-based framework enables the generation of continuous surface approximations, facilitating precise convergence analysis for Dirichlet energy and $L^\infty$ error evaluation in mean curvature. This methodology not only assures reliable surface approximation but also enables effective exploration of multiple minimal surfaces sharing the same boundary by employing varied initial conditions. Through this approach, we examine the dependence of solution multiplicity on boundary geometry, presenting a comprehensive analysis of the relationship between boundary contours and corresponding minimal surfaces.
• 11:00 - 11:30 Petr Pauš (Czech Technical University): Numerical simulation of dislocation multiple cross-slip
  Abstract: Our contribution deals with the phenomenon in material science called multiple cross-slip of dislocations in slip planes. The numerical model is based on a mean curvature flow equation with additional forcing terms included. The curve motion in 3D space is treated using our tilting method, i.e., mapping of a 3D situation onto a single plane where the curve motion is computed. The physical forces acting on a dislocation curve are evaluated in the 3D setting. This is a joint work with Michal Beneš and Miroslav Kolář.
• 11:30 - 12:00 John Sebastian Simon (Johann Radon Institute for Computation and Applied Mathematics): Robust shape optimization for missing Dirichlet data
  Abstract: A shape optimization problem subject to an elliptic equation that takes into account a missing data on the Dirichlet boundary condition is considered. The shape optimization problem is formulated by looking for the optimal deformation field that varies the domain where the Poisson equation is posed. The optimization problem is then recasted into the so- called low-regret and no-regret problems to obtain deformation fields which are robust in terms of the Dirichlet data. The analysis of the optimization problem were carried out, this includes providing appropriate topology for establishing the continuity of the deformation-to- state map and providing a proof establishing the existence of optimal deformation fields for both low-regret and no-regret problems. Due to the fact that the low-regret problem is a regularized version of the no-regret problem, convergence of the solutions to the former problem towards a solution of the latter was also provided. The first-order Gâteaux derivative of the reformulated objective function for the low-regret problem was also computed providing us a first-order necessary condition and a gradient-method based on Barzilai– Borwein to solve the problem numerically. Numerical examples are provided to illustrate the convergence previously mentioned, and to illustrate the robustness of the low-regret deformation fields with respect to the missing data.
• 12:00 - 14:00 Lunch break
• 14:00 - 14:30 Matteo Caggio (Institute of Mathematics, CAS): Vanishing viscosity limit for compressible fluids
  Abstract: We will present recent results and work in progress that concern the vanishing viscosity limit for compressible fluids. A density dependent viscosity model for a barotropic fluid and an heat-conductive model for a quasi-perfect gas will be considered.
• 14:30 - 15:00 Ivan Gudoshvnikov (Institute of Mathematcs, CAS): Regularity lost: why functions are not enough in continuum plasticity (so that we need measures).
  Abstract: The connection between plasticity phenomenon in mechanics and the sweeping process was noticed since the very discovery of the sweeping process as a mathematical problem. Using the ideas of J.-J. Moreau, we will discuss how the sweeping process can model the evolution of stresses in an elastoplastic medium. However, with the seemingly simplest constitutive law of ``perfect plasticty'' it is, in generall, impossible to find the strain rate as an $L_2$-function. We will demostrate this ill-posedness using Moreau's framework and also show that various constraint qualifications distinguish this case from the well-posed cases (spatially discrete models, plasticty with hardening).
• 15:00 - 15:30 Coffee break
• 15:30 - 15:40 Yuki Karasawa (Kanazawa University): A finite element scheme for a viscoelastic fluid flow
  Abstract: Resin products, which are numerous in our surroundings, are increasingly used in automobiles and airplanes due to their low density and lightweight. It is known that when melted, resins as raw materials behave as viscoelastic fluids, which are different from ordinary viscous fluids. Therefore, in numerical simulations related to polymer processing, it is desired to treat resins as viscoelastic fluids. However, numerical simulation methods for viscoelastic fluids are still under development. Viscoelastic fluid models exist called Oldroyd- B model, PTT, Giesekus, and FENE-P. All of these models include a characteristic term called the upper convection derivative, and the discretization of this derivative is important. A second-order accuracy approximation of the upper convection derivative in time based on the Lie derivative has been proposed and implemented in the finite difference framework [D.D. Medeiros, H. Notsu and C.M. Oishi. SIAM J. Numer. Anal., 59, 2955-2988, 2021]. In this talk, we propose a finite element scheme of second-order in time.
• 15:40 - 15:50 Ryuhei Wakida (Kanazawa University): A phase field model for fault rupture with dynamic elasticity equation
  Abstract: In Japan, earthquakes often occur, and they sometimes cause heavy damage. An earthquake can be roughly divided into two types, fault-type and trench-type. The purpose of our research is to reproduce a fault-type earthquake numerically. This study is based on a phase field crack growth model, and we expand the model to a wave equation. Furthermore, we will try to artificially introduce a friction term that acts on the fault plane into the phase field model. Adding a friction term to the fracture phase field model is challenging. For the finite element simulation, we use FreeFEM software. In addition, we introduce the parallel computing technique on FreeFEM, applying the overlapping Schwarz domain decomposition method and RAS preconditioner.
• 15:50 - 16:00 Hisanori Miyata (Kanazawa University): Analysis of mass conservation and gelation in Smoluchowski coagulation equation
  Abstract: This is a presentation of my research on the Smoluchowski coagulation equation. The equation is sometimes observed to show a gelation phenomenon, where mass is no longer conserved at a point in time, and the conditions for this will be discussed in this research. The first of the main theorems shows in mathematical terms, using the method of moments, the conditions under which mass is conserved in weak solutions of the equation. The other is conversely, showing the conditions for gelation.
• 16:00 - 16:10 Takeshi Nishikawa (Kanazawa University): Eigenvalue problems of the biharmonic equation with mixed boundary conditions
  Abstract: It is known that serious accidents can occur due to the eigen modes of an object, such as the collapse of the Tacoma Bridge in the United States in 1940. Therefore, the purpose of this study is to investigate the eigen modes of an object. For simplicity, we investigated the eigen vibrations of a plate in a two dimensional domain. This problem can be reduced to an eigenvalue problem of the biharmonic equation with a mixed boundary condition. To achieve this goal, we proposed a stable numerical computation method using the finite element method for the biharmonic equation. As a result, we were able to investigate how the plate vibrates through numerical calculation.
• 16:10 - 16:30 Final discussion
• 18:00 Dinner

Organizer