An Efficient Rescaling Algorithm for Simulating the Evolution of Multiple Elastically Stressed Precipitates

In this paper,we propose a space-time rescaling scheme for computing the long time evolution of multiple precipitates in an elastically stressed medium.The algorithm is second order accurate in time,spectrally accurate in space and enables one to simulate the evolution of precipitates in a fraction of the time normally used by fixed-frame algorithms.In particular,we extend the algorithm recently developed for single particle by Li et al.(Li,Lowengrub and Leo,J.Comput.Phys.,335(2007),554)to the multiple particle case,which involves key differences in the method.Our results show that without elasticity there are successive tip splitting phenomena accompanied by the formation of narrow channels between the precipitates.In presence of applied elastic field,the precipitates form dendrite-like structures with the primary arms aligned in the principal directions of the elastic field.We demonstrate that when the far-field flux decreaseswith the effective radius of the system,tip-splitting and dendrite formation can be suppressed,as in the one particle case.Depending on the initial position of the precipitates,we further observe that some precipitates grow while others may shrink,even when a positive far field flux is applied.

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions.The governing equations are solved numerically using a boundary integral method(BIM).A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications.In this paper,we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions(2D).We present a novel source dividing strategy to parallelize the treecode.Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors.This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution.We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media,which would be extremely difficult with a direct summation method due to CPU time constraint.The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.

A Fourth-Order Kernel-Free Boundary Integral Method for Interface Problems

This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.

Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell

We investigate the nonlinear dynamics of amoving interface in aHele-Shaw cell subject to an in-plane applied electric field.We develop a spectrally accurate numerical method for solving a coupled integral equation system.Although the stiffness due to the high order spatial derivatives can be removed using a small scale decomposition technique,the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands.We remove this physically imposed stiffness by employing a rescaling scheme,which accelerates the slow dynamics and reduces the computational cost.Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns.On the other hand,negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region.When no fluid is injected,and a negative current is utilized,the interface tends to approach the origin and break up into several drops.We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law(t∗−t)b,where t∗is the estimated pinch-off time.