Spectral Element Algorithm Research Articles

NUMERICAL SIMULATION OF MIXED ELECTROOSMOTIC/PRESSURE DRIVEN MICROFLOWS

A spectral element algorithm is developed to analyze mixed electroosmotic/pressure driven flows in complex two-dimensional geometries. The new algorithm exhibits spectral accuracy in resolving thin electric double layers. Mixed electroosmotic/pressure driven flows are simulated in straight channels. Electrokinetic pumping and the means of producing large pressure gradients in microchannels are explored. Finally, electroosmotic flow in a T-junction geometry is analyzed under various external electric field strengths. Flowrate in the T-junction is shown to vary linearly with the electroosmotic body forces in the Stokes flow regime.

An Unstructured hp Finite-Element Scheme for Fluid Flow and Heat Transfer in Moving Domains

This paper presents the formulation, validation, and application of a spectral/hp algorithm for numerical solution of the Navier–Stokes and heat/scalar transport equations in arbitrarily moving complex geometries. The new spectral element algorithm is based on the arbitrary Lagrangian Eulerian formulation and utilizes modal expansion of Jacobi polynomials in mixed triangular and quadrilateral elements in two dimensions. Time integration is performed by stiffly stable schemes and can deliver up to third-order time accuracy for transient problems. Test cases with sufficiently smooth solutions demonstrate spectral convergence and low phase/dissipation error. Preliminary simulation results for micro-mixers and micro-heat spreaders are also presented.

Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.

Numerical simulation of interface waves by high-order spectral modeling techniques

Few problems in elastodynamics have a closed-form analytical solution. The others can be investigated with semianalytical methods, but in general one is not sure whether these methods give reliable solutions. The same happens with numerical techniques: for instance, finite difference methods solve, in principle, any complex problem, including those with arbitrary inhomogeneities and boundary conditions. However, there is no way to verify the quantitative correctness of the solutions. The major problems are stability with respect to material properties, numerical dispersion, and the treatment of boundary conditions. In practice, these problems may produce inaccurate solutions. In this paper, the study of complex problems with two different numerical grid techniques in order to cross-check the solutions is proposed. Interface waves, in particular, are emphasized, since they pose the major difficulties due to the need to implement boundary conditions. The first method is based on global differential operators where the solution is expanded in terms of the Fourier basis and Chebyshev polynomials, while the second is the spectral element method, an extension of the finite element method that uses Chebyshev polynomials as interpolating functions. Both methods have spectral accuracy up to approximately the Nyquist wave number of the grid. Moreover, both methods implement the boundary conditions in a natural way, particularly the spectral element algorithm. We first solve Lamb’s problem and compare numerical and analytical solutions; then, the problem of dispersed Rayleigh waves, and finally, the two-quarter space problem. We show that the modeling algorithms correctly reproduce the analytical solutions and yield a perfect matching when these solutions do not exist. The combined modeling techniques provide a powerful tool for solving complex problems in elastodynamics.