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Plenary speakers
Plenary speakers
Leszek Demkowicz, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, TX, USA
Leszek Demkowicz, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, TX, USA
THE DISCONTINUOUS PETROV GALERKIN METHOD WITH OPTIMAL TEST FUNCTONS
THE DISCONTINUOUS PETROV GALERKIN METHOD WITH OPTIMAL TEST FUNCTONS
We are nearing a 10th anniversary of the fundamental idea of optimal test functions [1, 2]. With ten Ph.D. dissertations, over 50 papers and three international workshops on the subject, we have reached a certain degree of understanding and maturity but plenty of questions and research subjects remain open [3].
We are nearing a 10th anniversary of the fundamental idea of optimal test functions [1, 2]. With ten Ph.D. dissertations, over 50 papers and three international workshops on the subject, we have reached a certain degree of understanding and maturity but plenty of questions and research subjects remain open [3].
The lecture will present a short tutorial on the DPG method. The method combines two fundamental ideas: a Petrov-Galerkin discretization with optimal test functions, and variational formulations with broken (product) test spaces. We will discuss the equivalent minimum residual and mixed method formulations, and approximation of optimal test functions leading to the "Practical DPG method''.
The lecture will present a short tutorial on the DPG method. The method combines two fundamental ideas: a Petrov-Galerkin discretization with optimal test functions, and variational formulations with broken (product) test spaces. We will discuss the equivalent minimum residual and mixed method formulations, and approximation of optimal test functions leading to the "Practical DPG method''.
The presentation will include numerous numerical examples ranging from model problems in elasticity, Maxwell equations, acoustics to complex problems in fluid dynamics. We recommend reading reading the Encyclopedia article [3] before the lecture.
The presentation will include numerous numerical examples ranging from model problems in elasticity, Maxwell equations, acoustics to complex problems in fluid dynamics. We recommend reading reading the Encyclopedia article [3] before the lecture.
[1] L. Demkowicz and J. Gopalakrishnan, "A Class of Discontinuous Petrov-Galerkin Methods. Part I: The Transport Equation'', Comput. Methods Appl. Mech. Engrg. 199 (23-24): 1558-1572, 2010, see also ICES Report 2009-12.
[1] L. Demkowicz and J. Gopalakrishnan, "A Class of Discontinuous Petrov-Galerkin Methods. Part I: The Transport Equation'', Comput. Methods Appl. Mech. Engrg. 199 (23-24): 1558-1572, 2010, see also ICES Report 2009-12.
[2] L. Demkowicz and J. Gopalakrishnan, "A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions'', Numer. Meth. Part. D. E., 27:70-105, 2011. See also ICES Report 2009-16.
[2] L. Demkowicz and J. Gopalakrishnan, "A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions'', Numer. Meth. Part. D. E., 27:70-105, 2011. See also ICES Report 2009-16.
[3] L. Demkowicz and J. Gopalakrishnan, "Discontinuous Petrov-Galerkin (DPG) Method, Encyclopedia of Computational Mechanics, Second Edition, Eds. Erwin Stein, Rene de Borst, Thomas J. R. Hughes, Wiley 2018. See also ICES Report 2015/20.
[3] L. Demkowicz and J. Gopalakrishnan, "Discontinuous Petrov-Galerkin (DPG) Method, Encyclopedia of Computational Mechanics, Second Edition, Eds. Erwin Stein, Rene de Borst, Thomas J. R. Hughes, Wiley 2018. See also ICES Report 2015/20.
Irena Lasiecka, University of Memphis, Memphis, TN, USA
Irena Lasiecka, University of Memphis, Memphis, TN, USA
FEEDBACK SYNTHESIS AND NONSTANDARD RICCATI EQUATIONS ARISING IN OPTIMIZATION OF ULTRASOUND WAVES WITH HIGH INTNESITY FOCUS
FEEDBACK SYNTHESIS AND NONSTANDARD RICCATI EQUATIONS ARISING IN OPTIMIZATION OF ULTRASOUND WAVES WITH HIGH INTNESITY FOCUS
We will discuss boundary feedback control associated with PDE models arising in HIFU modles - which are PDE's of third order in time. This leads to a notion of a non-standard Riccati equations which provide suitable gain operators for the feedback control. Singularity of the control action compromises the usual regularity of the associated Riccati operators - making the analysis challenging particularly in the case of boundary controls. In this latter case, the loss of regularity is "double" - due to singularity caused by the appearance if time derivatives in control function and also due to the intrinsic loss associated with unbounded and un-closeable trace operators. In order to construct viable theory one needs to develop suitable regularity theory within the framework of non-smooth PDE optimization.
We will discuss boundary feedback control associated with PDE models arising in HIFU modles - which are PDE's of third order in time. This leads to a notion of a non-standard Riccati equations which provide suitable gain operators for the feedback control. Singularity of the control action compromises the usual regularity of the associated Riccati operators - making the analysis challenging particularly in the case of boundary controls. In this latter case, the loss of regularity is "double" - due to singularity caused by the appearance if time derivatives in control function and also due to the intrinsic loss associated with unbounded and un-closeable trace operators. In order to construct viable theory one needs to develop suitable regularity theory within the framework of non-smooth PDE optimization.
Boris Mordukhovich, Wayne State University, Detroit, MI, USA
Boris Mordukhovich, Wayne State University, Detroit, MI, USA
OPTIMIZATION AND FINITE-DIFFERENCE APPROXIMATIONS OF CONTROLLED SWEEPING PROCESSES WITH SOME APPLICATIONS
OPTIMIZATION AND FINITE-DIFFERENCE APPROXIMATIONS OF CONTROLLED SWEEPING PROCESSES WITH SOME APPLICATIONS
This talk addresses new classes of optimal control problems governed by convex and nonconvex versions of the sweeping/Moreau processes. Such control systems are described by discontinuous differential inclusions with intrinsic state constraints, which makes them highly challenging in control theory and applications. We develop the method of finite-difference/discrete approximations to deal with problems of this type. Besides some numerical advantages, it allows us to derive necessary optimality conditions for sweeping optimal solutions and then to proceed with applications to some practical dynamical models including the controlled planar crowd motion model of traffic equilibria.
This talk addresses new classes of optimal control problems governed by convex and nonconvex versions of the sweeping/Moreau processes. Such control systems are described by discontinuous differential inclusions with intrinsic state constraints, which makes them highly challenging in control theory and applications. We develop the method of finite-difference/discrete approximations to deal with problems of this type. Besides some numerical advantages, it allows us to derive necessary optimality conditions for sweeping optimal solutions and then to proceed with applications to some practical dynamical models including the controlled planar crowd motion model of traffic equilibria.
Patrizio Neff, Universität Duisburg-Essen, Essen, Germany
Patrizio Neff, Universität Duisburg-Essen, Essen, Germany
DESCRIBING METAMATERIALS WITH GENERALIZED CONTINUUM MODELS: THE RELAXED MICROMORPHIC APPROACH
DESCRIBING METAMATERIALS WITH GENERALIZED CONTINUUM MODELS: THE RELAXED MICROMORPHIC APPROACH
Metamaterials are man-made structures which may exhibit extraordinary features like band-gaps. In this talk I present the relaxed micromorphic continuum model and I show how that model may be used to capture the intricate response on a homogenized level. Decisive is the a priori identification of appearing material parameters based on knowledge of the underlying microstructure. The talk will be a tour d'horizon comprising modeling, analysis and verification against in-silico experiments.
Metamaterials are man-made structures which may exhibit extraordinary features like band-gaps. In this talk I present the relaxed micromorphic continuum model and I show how that model may be used to capture the intricate response on a homogenized level. Decisive is the a priori identification of appearing material parameters based on knowledge of the underlying microstructure. The talk will be a tour d'horizon comprising modeling, analysis and verification against in-silico experiments.
Daya Reddy, University of Cape Town, South Africa
Daya Reddy, University of Cape Town, South Africa
ANALYTICAL AND NUMERICAL INVESTIGATIONS OF LOCKING IN TRANSVERSELY ISOTROPIC ELASTICITY
ANALYTICAL AND NUMERICAL INVESTIGATIONS OF LOCKING IN TRANSVERSELY ISOTROPIC ELASTICITY
For isotropic materials the concept of volumetric locking in the context of low-order finite element approximations is well understood, and a variety of effective remedies exist: for example, the use of mixed methods, discontinuous Galerkin (DG) methods, or selective underintegration. Corresponding studies have been carried out, to a limited extent, to determine conditions under which locking related to inextensibility occurs, in small- and large-deformation contexts (see [1] and the references therein). The models treated in these works are of an isotropic material, with inextensibility imposed as a constraint. The present work is concerned with transversely isotropic linear elastic materials, which are characterized by 5 material parameters. The behaviour under limiting conditions of near-incompressibility and near-inextensibility are investigated. It is shown both through numerical examples and an analysis of finite element approximations that locking behaviour for low-order elements depends critically on the degree of anisotropy of the material, that is, on the ratio of Young’s moduli and Poisson ratios for the directions parallel and transverse to the direction characterizing transverse isotropy. In addition to conforming finite element approximations, DG approximations are also pursued, and conditions for their uniform convergence established, by extending analyses carried out for the isotropic problem (see for example [2, 3]).
For isotropic materials the concept of volumetric locking in the context of low-order finite element approximations is well understood, and a variety of effective remedies exist: for example, the use of mixed methods, discontinuous Galerkin (DG) methods, or selective underintegration. Corresponding studies have been carried out, to a limited extent, to determine conditions under which locking related to inextensibility occurs, in small- and large-deformation contexts (see [1] and the references therein). The models treated in these works are of an isotropic material, with inextensibility imposed as a constraint. The present work is concerned with transversely isotropic linear elastic materials, which are characterized by 5 material parameters. The behaviour under limiting conditions of near-incompressibility and near-inextensibility are investigated. It is shown both through numerical examples and an analysis of finite element approximations that locking behaviour for low-order elements depends critically on the degree of anisotropy of the material, that is, on the ratio of Young’s moduli and Poisson ratios for the directions parallel and transverse to the direction characterizing transverse isotropy. In addition to conforming finite element approximations, DG approximations are also pursued, and conditions for their uniform convergence established, by extending analyses carried out for the isotropic problem (see for example [2, 3]).
[1] Wriggers P., Schröder J. and Auricchio F. Finite element formulations for large strain anisotropic material with inextensible fibers. Adv Model Simul Eng Sci (2016) 3(1):25.
[1] Wriggers P., Schröder J. and Auricchio F. Finite element formulations for large strain anisotropic material with inextensible fibers. Adv Model Simul Eng Sci (2016) 3(1):25.
[2] Wihler, T. Locking-free dgfem for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45–75.
[2] Wihler, T. Locking-free dgfem for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45–75.
[3] Grieshaber B.J., McBride A.T. and Reddy B.D. Uniformly convergent interior penalty methods using multilinear approximations, for problems in elasticity. SIAM J. Numer. Anal. 53 (2015) 2255–2278.
[3] Grieshaber B.J., McBride A.T. and Reddy B.D. Uniformly convergent interior penalty methods using multilinear approximations, for problems in elasticity. SIAM J. Numer. Anal. 53 (2015) 2255–2278.
Meir Shillor, Oakland University, Rochester, MI, USA
Meir Shillor, Oakland University, Rochester, MI, USA
MODELS, ANALYSIS AND SIMULATIONS IN CONTACT MECHANICS AND EPIDEMIOLOGY
MODELS, ANALYSIS AND SIMULATIONS IN CONTACT MECHANICS AND EPIDEMIOLOGY
The talk presents an overview of a few new results in the Mathematical Theory of Contact Mechanics (MTCM). In particular those related to modeling and analysis of contact, friction, adhesion and wear. Then, it presents a few new results in the applications of the theory of dynamical systems to modeling of the spread of disease.
The talk presents an overview of a few new results in the Mathematical Theory of Contact Mechanics (MTCM). In particular those related to modeling and analysis of contact, friction, adhesion and wear. Then, it presents a few new results in the applications of the theory of dynamical systems to modeling of the spread of disease.
In the first part I will present: more general models for dynamic or quasistatic contact; some thoughts on the coefficient of friction and the transition from a stick state to sliding; frictional contact problems in thermistors, which are systems where thermal and electrical effects are included; and wear of surfaces in contact and the diffusion of the wear particles on the contacting surfaces. Then, I will present a new general existence theorem for quasistatic processes that include such models, and allows for some of the system coefficients to have random components. This is of considerable practical importance, since too often various system parameters can only be estimated and are not know precisely. In addition, there is some indication that the coefficient of friction may be best described as a random selection from a statistical distribution.
In the first part I will present: more general models for dynamic or quasistatic contact; some thoughts on the coefficient of friction and the transition from a stick state to sliding; frictional contact problems in thermistors, which are systems where thermal and electrical effects are included; and wear of surfaces in contact and the diffusion of the wear particles on the contacting surfaces. Then, I will present a new general existence theorem for quasistatic processes that include such models, and allows for some of the system coefficients to have random components. This is of considerable practical importance, since too often various system parameters can only be estimated and are not know precisely. In addition, there is some indication that the coefficient of friction may be best described as a random selection from a statistical distribution.
The second part is dedicated to models for the Middle East Respiratory Syndrome (MERS), which is an epidemic currently active in the middle east. Then, I will describe models, their analysis, and simulations for the spread of Chagas disease in Central and South America.
The second part is dedicated to models for the Middle East Respiratory Syndrome (MERS), which is an epidemic currently active in the middle east. Then, I will describe models, their analysis, and simulations for the spread of Chagas disease in Central and South America.
Mircea Sofonea, University of Perpignan Via Domitia, France
Mircea Sofonea, University of Perpignan Via Domitia, France
HISTORY-DEPENDENT VARIATIONAL-HEMIVARIATIONAL INEQUALITIES IN CONTACT MECHANICS
HISTORY-DEPENDENT VARIATIONAL-HEMIVARIATIONAL INEQUALITIES IN CONTACT MECHANICS
We present recent results in the study of variational-hemivariational inequalities with applications to Contact Mechanics. We start by introducing the concept of history-dependent operator together with relevant examples in analysis, ordinary differential equations and mechanics. Then, we state and prove an existence and uniqueness result for a class of variational-hemivariational inequalities with history-dependent operators, the so-called history-dependent inequalities. The proof is based on arguments of pseudomonotonicity and fixed point. Under additional assumptions, we proceed with the study of the behavior of the solution with respect to the set of constraints and prove a continuous dependence result. To this end we use various estimates, monotonicity arguments and the properties of the Clarke subdifferential.
We present recent results in the study of variational-hemivariational inequalities with applications to Contact Mechanics. We start by introducing the concept of history-dependent operator together with relevant examples in analysis, ordinary differential equations and mechanics. Then, we state and prove an existence and uniqueness result for a class of variational-hemivariational inequalities with history-dependent operators, the so-called history-dependent inequalities. The proof is based on arguments of pseudomonotonicity and fixed point. Under additional assumptions, we proceed with the study of the behavior of the solution with respect to the set of constraints and prove a continuous dependence result. To this end we use various estimates, monotonicity arguments and the properties of the Clarke subdifferential.
Next, we consider a mathematical model which describes the equilibrium of a locking material with memory, in contact with an obstacle. We comment the model and state its weak formulation, which is in a form of a history-dependent variational-hemivariational inequality for the displacement field. We use our abstract results to prove the unique weak solvability of the model and the continuous dependence of the solution with respect to the set of constraints. We apply this convergence result in the study of an optimization problem associated to the contact model. Finally, we list additional results concerning the numerical analysis of history-dependent variational-hemivariational inequalities and provide numerical simulations in the study of a second mathematical model which describes the contact between a viscoelastic body and a rigid-deformable foundation.
Next, we consider a mathematical model which describes the equilibrium of a locking material with memory, in contact with an obstacle. We comment the model and state its weak formulation, which is in a form of a history-dependent variational-hemivariational inequality for the displacement field. We use our abstract results to prove the unique weak solvability of the model and the continuous dependence of the solution with respect to the set of constraints. We apply this convergence result in the study of an optimization problem associated to the contact model. Finally, we list additional results concerning the numerical analysis of history-dependent variational-hemivariational inequalities and provide numerical simulations in the study of a second mathematical model which describes the contact between a viscoelastic body and a rigid-deformable foundation.
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