We investigate topology optimization based on the SIMP method on CUDA enabled video cards in three dimensions. Using finite elements, linear elasticity is solved by a matrix-free conjugate gradient method entirely on the GPU. Due to the unique requirements of the SIMD stream processors, special attention is given to procedural generation of matrix vector products entirely on the graphics card. ...

Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic partial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equations, we give a complete solution, and for others, we extend previously known results. For example, we determine under which conditions two cons...

The physical world is traditionally organized into various systems: electromagnetism, perfect fluids, Klein-Gordon fields, elastic media, gravitation, etc. Our descriptions of these individual systems have certain features in common: Use of fields on a fixed space-time manifold M , a geometrical interpretation of the fields in terms of M , partial differential equations on these fields, an init...

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L-distance of the gradients of the state functions....

In this paper we consider several moving mesh partial diierential equations which are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh equations which have been used by others. An analysis of their stability is done. It is seen that a key term for most of these moving mesh PDEs is a source-like term which measures the level of equi...

These are my incomplete lecture notes for the graduate introduction to PDE at Brown University in Fall 2005. The lectures on Laplace’s equation and the heat equation are included here. Typing took too much work after that. I hope what is here is still useful. Andreas Klöckner’s transcript of the remaining lectures is also posted on my website. Those however have not been proofread. Comments are...

In many problems related to cryopreservation of living cells and tissues, the formation of ice in intra and extracellular liquids plays very important role. There are many models predicting the homogeneous ice nucleation an formation of dendrites. Nevertheless, the application of such models requires fitting parameters and verifications trough measurements of the ice content during freezing. Th...

A partial differential equation (PDE) of order m is a relation of the form F (x, u,Du,Du, · · · , Du) = 0. (0.1) Here F is a given function of x ∈ R, ”unknown” function u = u(x), and its derivatives up to order m. We denote Du the set of all the derivatives of u of order k. Using multi-indices l = (l1, · · · , ln), i.e. vectors in R with nonnegative integer components, we can write Du = { Du = ...