Christof Eck

A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies

This article presents a fast and accurate adaptive algorithm that numerically solves a two-scale model with continuous inter-scale dependencies. The examined sample two-scale model describes a phase transition of a binary mixture with the evolution of equiaxed dendritic microstructures. It consists of a macroscopic heat equation and a family of microscopic cell problems that model the phase transition of the mixture. Both scales are coupled: the macroscopic temperature field influences the evolution of the microstructure and the microscopic fields enter to the macroscopic heat equation via averaged coefficients. Adaptivity exploits the constitutive assumption that the evolving microstructure depends in a continuous way on the macroscopic temperature field: macroscopic nodes with similar temperature evolutions use the same microscopic data. A suitable metric compares temperature evolutions and adaptive methods select active macroscopic nodes. Microscopic cell problems are solved for active nodes only; microscopic data in inactive nodes is approximated from microscopic data of active nodes with a similar temperature evolution. The set of active nodes is updated in course of the simulation: active nodes are deactivated until all active nodes have unsimilar temperature evolutions, and inactive nodes are activated until for every inactive node there exists at least one active node with a similar temperature evolution. Numerical examples, in two and in three space dimensions, show that the adaptive solution is only slightly less accurate than the direct solution, but it is computationally much more efficient. Therefore, the adaptive algorithm enables the solution of two-scale models with continuous inter-scale dependencies on large computational macroscopic and microscopic grids within an acceptable period of time for computation.

Multiscale Problems in Solidification Processes

Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy. Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale β related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in β. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error.

Error Estimates for a Finite Element Discretization of a Phase Field Model for Mixtures

We derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures. Special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region. The main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system, a suitable regularity of the exact solutions, and a spectrum estimate for the operator of the Allen-Cahn equation. The spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model. This is done by a technique introduced in [X. Feng and A. Prohl, Math. Comp., 73 (2004), pp. 541-567] for phase transitions of pure materials with linear constitutive relations.

Existence of Solutions to a Nonlinear Coupled Thermo-Viscoelastic Contact Problem with Small Coulomb Friction

The solvability of a coupled thermoviscoelastic contact problem with Coulomb friction is investigated. The heat generated by friction is described by a boundary term of quadratic order. The tensor of thermal conductivity is dependent on the temperature gradient and satisfies a certain growth condition.

Basic Principles of Thermodynamics

Thermodynamics is concerned with the definition and relation of notions like temperature, pressure and volume, with the first law of thermodynamics as a basis. The additional notion of entropy is characterized by the second law of thermodynamics. Temperature can be viewed as an integrating factor. Equilibrium conditions can be formulated by means of thermodynamic potentials. Legendre transforms and differential forms are introduced as auxiliary means. The theory is extended to mixtures to include chemical reactions, i.e., the mass action law in equilibrium and also kinetic reactions.

Ordinary Differential Equations

One-dimensional oscillations are described by linear ordinary differential equations with closed form solutions extendable to forced oscillations. Besides the Lagrangian the Hamiltonian form of mechanics is introduced. Further applications are the motion of space frames (from Chapter 3) and two species models from population dynamics. By means of qualitative analysis (using phase portraits) the long time behavior can be analyzed. The principle of linearized stability reduces the stability question to linear systems, where it can be completely studied. Finally also variational problems and optimal control problems are discussed.

Free Boundary Problems

Obstacle and contact problems leading to variational inequalities as well as models for porous media are studied. A main theme is phase transitions appearing in the context of solidification processes. Finally, also free boundary problems in fluid dynamics are considered.

Partial Differential Equations

Elements of the theory of partial differential equations that are relevant for the modeling process are discussed. Elliptic, parabolic, hyperbolic equations as well as the Navier-Stokes and the Euler equations are treated. Finally, boundary layer theory is discussed.

Systems of Linear Equations

Electrical networks and space frames in its stationary state can be described by sets of linear equations of similar structure. This is extendable to alternating current circuits by the use of complex parameters. The set of equations can be written in a form equivalent to a constrained optimization problem, giving rise to a dual form by means of Lagrange multipliers.

Probleme mit freiem Rand

Viele Anwendungen in Naturwissenschaften und Technik fuhren auf Problemstellungen, bei denen die Geometrie des Gebietes, auf dem eine Gleichung gelost werden soll, a priori unbestimmt ist. Ist eine partielle Differentialgleichung in einem Gebiet zu losen, von dem ein Teil des Randes unbekannt ist, so spricht man von einem Problem mit freiem Rand. Zusatzlich zu den ublichen Randbedingungen, die gebraucht werden, um die partielle Differentialgleichung zu losen, sind in diesem Fall weitere Bedingungen am freien Rand zu stellen. Probleme mit freiem Rand tauchen unter anderem bei folgenden Fragestellungen auf: Schmelz- und Erstarrungsphanomene (Stefan–Problem), Hindernisprobleme fur elastische Membranen, Kontaktprobleme bei elastischen Verformungen, Wachstum von Tumoren, Stromungen mit freien Oberflachen und Bewertung von Finanzderivaten.