Introduction to linear and nonlinear programming

Rigid-body dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigid-body dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painleve [C. R. Acad. Sci. Paris, 121 (1895), pp. 112--115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. 
The new mathematical developments in rigid-body dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.-L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lotstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigid-body dynamics with Coulomb friction and impulses.

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Rigid-Body Dynamics with Friction and Impact

Rate-independent systems occur as limit problems in many physical and mechanical problems, if the interesting time scales are much longer than the intrinsic time scales in the system. Rate-independent systems are sometimes also called “quasistatic systems,” however, the term “quasistatic” is often used in a more general sense—namely, if the inertial terms in a system are neglected, but viscous effects might still be present. The chapter discusses different aspects of rate-independent models or hysteresis operators in the context of continuum mechanics. There are several areas in these fields that have evolved quite independently and have developed their own languages and notation. The chapter compares these different approaches by translating them into one language, and thus provides a useful overview of the different methods in the field.

Chapter 6 - Evolution of Rate-Independent Systems

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation, we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

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Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

PREFACE INTRODUCTION Notations Linear Elasticity Formulation of Contact Problems Variational Principles in Mechanics The Static Contact Problem Geometry of Domains The Method of Tangential Translations BACKGROUND Fixed Point Theorems Some General Remarks Crash Course in Interpolation Besov and Lizorkin-Triebel Spaces The Potential Spaces Vector-Valued Sobolev and Besov Spaces Extensions and Traces Spaces on Domains STATIC AND QUASISTATIC CONTACT PROBLEMS Coercive Static Case Semicoercive Contact Problem Contact Problems for Two Bodies Quasistatic Contact Problem DYNAMIC CONTACT PROBLEMS A Short Survey About Results for Elastic Materials Results for Materials With Singular Memory Viscoelastic Membranes Problems With Given Friction DYNAMIC CONTACT PROBLEMS WITH COULOMB FRICTION Solvability of Frictional Contact Problems Anisotropic Material Isotropic Material Thermo-Viscoelastic Problems BIBLIOGRAPHY LIST OF NOTATION SUBJECT INDEX

Unilateral contact problems : variational methods and existence theorems

We prove the existence of solutions to the static contact problem with Coulomb friction, provided that the coefficient of friction is small enough. The proof employs the penalty method and a certain smoothing procedure for the friction functional. Using optimal trace estimates for the solutions of the Lame equations, we calculate an upper bound for the admissible coefficient of friction which is greater than the corresponding bounds proposed by Necas, Jarusek and Haslinger (1980) and by Jarusek (1983).

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Existence results for the static contact problem with coulomb friction

The existence of solutions to the dynamic contact problem with Coulomb friction for viscoelastic bodies is proved with the use of penalization and regularization methods. The contact condition, which describes the nonpenetrability of mass, is formulated in velocities. The coefficient of friction may depend on the solution but is assumed to be bounded by a certain constant.

Dynamic contact problems with small coulomb friction for viscoelastic bodies: existence of solutions

The term electrowetting is commonly used for phenomena where shape and wetting behavior of liquid droplets are changed by the application of electric fields. We develop and analyze a model for electrowetting that combines the Navier–Stokes system for fluid flow, a phase-field model of Cahn–Hilliard type for the movement of the interface, a charge transport equation, and the potential equation of electrostatics. The model is derived with the help of a variational principle due to Onsager and conservation laws. A modification of the model with the Stokes system instead of the Navier– Stokes system is also presented. The existence of weak solutions is proved for several cases in two and three space dimensions, either with non-degenerate or with degenerate electric conductivity vanishing in the droplet exterior. Some numerical examples in two space dimensions illustrate the applicability of the model. 2000 Mathematics Subject Classification: 35D05, 35D10, 35R35, 76M30.

https://www.ems-ph.org/journals/show_pdf.php?issn=1463-9963&vol=11&iss=2&rank=4

Christof Eck

Papers

Unilateral contact problems : variational methods and existence theorems

Existence results for the static contact problem with coulomb friction

Dynamic contact problems with small coulomb friction for viscoelastic bodies: existence of solutions

On a phase-ﬁeld model for electrowetting

A Two-Scale Method for the Computation of Solid Liquid Phase Transitions with Dendritic Microstructure