Time-varying Domain Research Articles

Invariant Solutions of Differential Games and Hamilton--Jacobi--Isaacs Equations for Time-Measurable Hamiltonians

We characterize invariance of time-varying domains with respect to differential games with time-measurable dynamics. We deduce from this result a new definition of viscosity solutions to some first order Hamilton--Jacobi equations with time-measurable Hamiltonians.

Boundary value problems for parabolic Lame systems in time-varying domains

Boundary value problems for parabolic Lame systems in time-varying domains

An adaptive discontinuous Galerkin method for viscoplastic analysis

This paper describes an h-adaptive, space-time discontinuous Galerkin finite element method for quasi-static viscoplastic response in a time-varying domain. The equations of equilibrium and the evolution equations in the viscoplastic material model comprise an elliptic/hyperbolic system. We focus here on two aspects of the model: stabilization of the hyperbolic subproblem and residual-based error estimates and adaptive algorithms for viscoplastic analysis.

The method of layer potentials for the heat equation in time-varying domains

Singular integrals The David buildup scheme Absolute continuity and Dirichlet-Neumann problems.

Summary of GENESIS work at the European Centre for Medium-range Weather Forecasts (ECMWF)

Benchmarks with simplified atmospheric models on small parallel computers, and performance estimates with operational weather models on massively parallel computers, indicate that atmospheric models can be efficiently implemented on future parallel supercomputers with thousands of processors. By using the so-called transposition strategy, which employs a time-varying domain decomposition, current sequential numerical methods, and even most subroutines, can be employed in the parallel models without any modifications.

Probability of plastic collapse of a structural system under nonstationary load processes

This paper deals with the reliability analysis of a complex structural system, in which the failure of the system is due to plastic collapse. Of particular interest herein is the computation of the time-variant probability of the structural system failure under nonstationary load processes. The theoretically most attractive approach for this type of problem in time-dependent structural reliability analysis is to employ the concept of first passage probability in stochastic process theory. Because the use of the generalized Rice formula poses some difficulty, the outcrossing problem will be formulated herein in another way, i.e., using the original Rice formula directly. This situation allows the multidimensional integral to be reduced to a one-dimensional integral. Also it can include smooth nonstationary processes as well as time-variant domain boundaries. An example is given to illustrate the method.

Motion planning in time-varying domains: the case of cyclic motions

Motion planning is studied in a time-varying environment. The environment contains a set of obstacles that have cyclic (i.e. repeated) motions. Cyclic motion is a generalization over simpler motion patterns such as straight-line motions and is capable of representing much more complex and realistic time-varying environments than those previously considered. In our model, the destination point is also allowed to have a cyclic motion. Two methods are presented for generating a collision-free path and their convergence property is established. The first method is suited for sensor-based navigation while the second method is an algorithm for gross motion planning. Computer simulation is presented to demonstrate the feasibility of our approach.

Finite element modeling of flow in a coal seam with underground coal gasification cavities

A two-dimensional, finite element groundwater flow model was developed to study the movement of water in a coal seam in which large cavities were created by underground coal gasification (UCG) burns. The burns extract natural gas from the coal without any environmental disturbance at the ground level. On completion of the burn, groundwater mixes with the products of combustion in the cavity and the pollutants move into the coal seam. Mathematical modeling is used to study the movement and fate of these pollutants with time. Water quality modeling has to be preceded by flow modeling to determine velocities as a function of time. This paper studies the flow aspects of the problem. A new code was developed because of the unique requirements of the model. The model utilizes linear triangles to discretize the coal seam in plan and takes into account unsteady flow, anisotropic media, internal boundary conditions imposed by the cavities and a time-varying domain of flow in plan. The model was applied to a series of UCG burns in Hanna, Wyoming. Estimates of the time of filling of the five cavities were obtained. Comparisons of measured and computed potential head are presented at different points in the coal seam. Flow modeling can thus be used to predict the movement of water into and out of UCG cavities while the output of velocities is necessary for water quality modeling.

Treatment of multi-dimensional moving boundary problems by coordinate transformation

A method based on coordinate transformation which transforms the time-varying domain into an invariant one, is given for solving multi-dimensional solidification/melting problems. The present method differs from its predecessors in that it transforms one space variable only while others remain unchanged. This results in tremendous amounts of saving in computations in comparison to methods proposed by earlier authors. Two sample problems are treated by the method in two dimensions one concerned with solidification of the square prism and the other with melting in a rectangular prism. The numerical results obtained by the present method are found to be in good agreement with those due to earlier authors. The method is further extended to three dimensions through a problem of cuboid solidification. Obtained in the present paper are perhaps the only available results for the complete solidification of the cuboid.