Abstract : The problem area addresses non-smooth problems stemming from mechanics and described by partial differential equations, inverse and ill-posed problems, non-smooth and nonconvex optimization, optimal control problems, multiscale analysis and homogenization, shape and topology optimization. We focus but are not limited to singularities like cracks, inclusions, aerofoils, defects and inhomogeneities arising in composite structures and multi-phase continua, which are governed by systems of variational equations and inequalities. The minisymposium objectives are directed toward sharing advances attained in the mathematical theory, numerical methods, and application of non-smooth problems.
Organizer(s) : Victor Kovtunenko, Hiromichi Itou, Alexander Khludnev, Evgeny Rudoy
[00222] Recent progress on the irreversible fracture phase field model
Format : Talk at Waseda University
Author(s) :
Masato Kimura (Kanazawa University)
Abstract : We would like to present our recent progress in the study of the fracture phase field model of the irreversible type. It not only enables us to simulate various kinds of crack propagation phenomena but also realizes a non-healing property and a natural energy gradient structure simultaneously.
[00267] Fractional Korn inequalities in bounded domains
Format : Talk at Waseda University
Author(s) :
Davit Harutyunyan (University of California Santa Barbara)
Hayk Mikayelyan (University of Nottingham Ningbo China)
Abstract :
The validity of Korn's first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn's first inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(\Omega)$ Sobolev fields in open and bounded $C^{1}$-regular domains $\Omega\subset \mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain $\Omega\subset \mathbb R^n$ we construct counterexamples to the inequality, i.e., Korn's first inequality fails to hold in bounded domains. The proof of the inequality in the case $ps>1$ follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [\textit{Commun. Math. Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in the case $ps<1$ are interpolations of a constant affine rigid motion inside the domain away from the boundary, and of the zero field close to the boundary.
[00266] On Phase Field Approach for Crack Propagation due to Water Pressure in Porous Medium
Format : Talk at Waseda University
Author(s) :
Sayahdin Alfat (Kanazawa University)
Masato Kimura (Kanazawa University)
Abstract : The crack propagation in the material due to water pressure was studied. This study involved the poroelasticity theory proposed by M. A. Biot. This study is divided into two parts. In the first part, we derived the poroelasticity theory, and its energy equality and presented several numerical examples. In the second part, we introduce our phase field model with unilateral contact conditions for desiccation cracking by coupling with the Biot model and show energy equality.
[00171] Asymptotic series solution of variational Stokes problems in planar domain with crack-like singularity
Format : Talk at Waseda University
Author(s) :
Victor Kovtunenko (University of Graz)
Abstract : Variational problems for incompressible fluids and solids descried by stationary Stokes equations in a planar domain with crack are considered. Based on the Fourier asymptotic analysis, general solutions are derived analytically as power series with respect to the distance to the crack tip. The logarithm terms and angular functions are accounted in the asymptotic expansion using recurrence relations. Boundary conditions of Dirichlet, Neumann, impermeability, non-penetration, and shear at the crack faces determine admissible exponents and parameters in the power series. The principal asymptotic terms are derived in the sector of angle 2π, which determine a square-root singularity at the crack tip and presence of log-oscillations of variational solutions for the Stokes problems.
[00288] Evolution equations with complete irreversibility
Format : Talk at Waseda University
Author(s) :
Goro Akagi (Tohoku University)
Abstract : In this talk, recent developments of studies on evolution equations with complete irreversibility, i.e., solutions are constrained to be monotone in time, will be reviewed. In particular, such evolution equations often arise from fracture and damage mechanics. From mathematical points of view, they are classified as fully nonlinear PDEs, and therefore, it is in general more difficult to prove their well-posedness and reveal dynamics of their solutions. However, by change of variables, they can be rewritten as doubly-nonlinear evolution equations, whose energy and variational structures are available for the analysis. Moreover, they are sometimes equivalently reformulated as semilinear obstacle problems whose obstacle functions coincide with initial data. It will also give us a clue for the analysis. In this talk, we shall overview recent results on diffusion equations and Allen-Cahn equations with complete irreversibility and also some phase-field systems arising from fracture models.
[00285] H^2-regularity up to boundary for a Bingham fluid model
Format : Talk at Waseda University
Author(s) :
Takahito Kashiwabara (The University of Tokyo)
Abstract : Bingham fluid is a model describing the motion of viscoplastic materials. The problem is formulated by a variational inequality, to which weak solvability in the Sobolev space H^1 is well known. However, regularity in H^2 up to the boundary, unlike its interior counterpart, seems to remain open. In this talk, we present such a result for the homogeneous slip boundary value problem, avoiding the difficulty of being unable to get good pressure estimates.
[00279] On a generalization Kelvin–Voigt model with pressure-dependent moduli
Format : Talk at Waseda University
Author(s) :
Hiromichi Itou (Tokyo University of Science)
Victor Kovtunenko (University of Graz)
Kumbakonam Rajagopal (Texas A&M University)
Abstract : In this talk, we discuss a generalization Kelvin–Voigt model of viscoelasticity whose material moduli depend on the pressure in which both the Cauchy stress and the linearized strain appear linearly. This model is derived from an implicit constitutive relation, and is well-suited to describe porous materials like concrete, ceramics. We show well-posedness for the corresponding variational problem by thresholding the moduli.
[00287] A reconstruction problem in nanoscale processing by transverse dynamic force microscopy
Format : Online Talk on Zoom
Author(s) :
Alemdar Hasanov Hasanoglu (Kocaeli University)
Abstract : In this study, the dynamic model of reconstruction of the shear force in the transverse dynamic force microscopy (TDFM)-cantilever tip-sample interaction is proposed. For this inverse problem, an input-output operator is introduced and then the compactness of this operator, thus, the ill-posedness of the inverse problems is proved. The least square solution of the inverse problem is introduced through the Tikhonov functional. The Lipschitz continuity of the input-output operator is proved. As a consequence of this, the existence of the least square solution is proved. An explicit formula for the Fr\'{e}chet gradient is derived by making use of the unique solution of the corresponding adjoint problem. This allows us to construct an effective and fast reconstruction algorithm, as the presented computational experiments show.
[00203] On Kirchhoff-Love plates with thin elastic junction
Format : Online Talk on Zoom
Author(s) :
Alexander Khludnev (Lavrentyev Institute of Hydrodynamics of RAS)
Abstract : The talk concerns an equilibrium problem for two elastic
plates connected by a thin junction (bridge) in a case of Neumann
boundary conditions, which provide a non-coercivity for the problem. An
existence of solutions is proved. Passages to limits are justified with respect
to the rigidity parameter of the junction. In particular, the rigidity
parameter tends to infinity and to zero. Limit models are investigated.
[00235] Optimal Location Problem for Heterogeneous Bodies with Separate and Joined Rigid Inclusions
Format : Online Talk on Zoom
Author(s) :
Niurgun Lazarev (North Eastern Federal University)
Abstract : Nonlinear mathematical models describing an equilibrium state of heterogeneous bodies which may come into contact with a fixed non-deformable obstacle are investigated. A possible mechanical interaction of the body and the obstacle is described with the help of the Signorini-type non-penetration condition. We suppose that the heterogeneous bodies consist of an elastic matrix and one or two built-in volume (bulk) rigid inclusions. One of the inclusions can vary its location along a given curve. Considering a location parameter as a control parameter, we formulate an optimal control problem with a cost functional specified by an arbitrary continuous functional on the solution space. Assuming that the location parameter varies in a given closed interval, the solvability of the optimal control problem is established. Furthermore, it is shown that the equilibrium problem for the heterogeneous body with joined two inclusions can be considered as a limiting problem for the family of equilibrium problems for heterogeneous bodies with two separate inclusions.
[00210] Multiscale analysis of stationary thermoelastic vibrations of a composite material
Format : Online Talk on Zoom
Author(s) :
Evgeny Rudoy (Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences)
Abstract : The stationary vibrations problem is studied for a planar thermoelastic body incorporating thin inclusions. This problem contains two small positive, which describe the thickness of an individual inclusion and the distance between two neighboring inclusions. Relying on the variational formulation, by means of methods of asymptotic analysis, we investigate the behavior of solutions as parameters tend to zero. We construct models corresponding to limit cases. The work is supported by Russian Scientific Foundation (№ 22-21-00627).
[00207] An impulsive pseudoparabolic equation with an infinitesimal transition layer
Format : Online Talk on Zoom
Author(s) :
Sergey Alexandrovich Sazhenkov (Altay State University, Barnaul)
Abstract : We study the two-dimensional Cauchy problem for the non-instantaneous impulsive pseudoparabolic equation. Such equations arise in filtration theory, thermodynamics, etc. We rigorously justify the passage to the instantaneous impulsive equation and show that, as the duration of the impulse tends to zero, the infinitesimal transition layer is formed, which inherits the profile of the original non-instantaneous impulsive impact. This is a joint work with Dr. Ivan Kuznetsov of the Lavrentyev Institute of Hydrodynamics, Russia.
