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  • 1
    Electronic Resource
    Electronic Resource
    Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations (1995)
    Springer
    Journal of scientific computing 10 (1995), S. 93-137 
    Details
    ISSN: 1573-7691
    Keywords: Domain decomposition ; nested dissection ; LU-factorization ; time integration ; Newton's method ; spectral elements ; finite elements
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02087962
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    An incomplete nested dissection algorithm for parallel direct solution of finite element discretizations of partial differential equations (1993)
    Springer
    Journal of scientific computing 8 (1993), S. 373-387 
    Details
    ISSN: 1573-7691
    Keywords: Domain decomposition ; nested dissection ; LU-factorization ; parallel computers ; MIMD
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract A multilevel algorithm is presented for direct, parallel factorization of the large sparse matrices that arise from finite element and spectral element discretization of elliptic partial differential equations. Incomplete nested dissection and domain decomposition are used to distribute the domain among the processors and to organize the matrix into sections in which pivoting is applied to stabilize the factorization of indefinite equation sets. The algorithm is highly parallel and memory efficient; the efficient use of sparsity in the matrix allows the solution of larger problems as the number of processors is increased, and minimizes computations as well as the number and volume of communications among the processors. The number of messages and the total volume of messages passed during factorization, which are used as measures of algorithm efficiency, are reduced significantly compared to other algorithms. Factorization times are low and speedups high for implementation on an Intel iPSC/860 hypercube computer. Furthermore, the timings for forward and back substitutions are more than an order-of-magnitude smaller than the matrix decomposition times.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01061145
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  • 3
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    Petrov-Galerkin methods for natural convection in directional solidification of binary alloys (1987)
    In:  Other Sources
    Details
    Publication Date: 2011-08-19
    Description: A Petrov-Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov-Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov-Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov-Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.
    Keywords: FLUID MECHANICS AND HEAT TRANSFER
    Type: International Journal for Numerical Methods in Fluids (ISSN 0271-2091); 7; 761-791
    Format: text
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  • 4
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    Third-order resonance effects and the nonlinear stability of drop oscillations (1987)
    In:  Other Sources
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    Publication Date: 2011-08-19
    Description: The three-dimensional nonlinear oscillations of an isolated, inviscid drop with surface tension are studied by a multiple timescale analysis and pre-averaging applied to the variational principle for the appropriate Lagrangian. Amplitude equations are derived which describe the generic cubic resonance caused by the spatial degeneracy of the eigenfrequencies of the linear normal modes. This resonant coupling leads to the instability of the finite amplitude axisymmetric oscillations to small nonaxisymmetric perturbations, as is demonstrated here for the three- and four-lobed normal modes. Solutions to the interaction equations that describe finite amplitude, nonaxisymmetric traveling-wave solutions are also obtained and their stability is investigated. A nongeneric cubic resonance between the two-lobed and four-lobed oscillatory modes leads to quasi-periodic motions.
    Keywords: FLUID MECHANICS AND HEAT TRANSFER
    Type: Journal of Fluid Mechanics (ISSN 0022-1120); 183; 95-121
    Format: text
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  • 5
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    Dynamic centering of liquid shells (1987)
    In:  Other Sources
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    Publication Date: 2019-07-12
    Description: The moderate-amplitude axisymmetric oscillations of an inviscid liquid shell surrounding an incompressible gas bubble are calculated by a multiple-time-scale expansion for initial deformations composed of two-lobed perturbations of the shell and a displacement of the bubble from the center of mass of the liquid. Two types of small-amplitude motion are identified and lead to very different nonlinear dynamic interactions, as described by the results valid up to second order in the amplitude of the initial deformation. In the 'bubble mode', the oscillations of the captive bubble and the liquid shell are exactly in phase and the bubble vibrates about its initial eccentric location. The bubble moves toward the center of the drop when the shell is perturbed into a 'sloshing mode' of oscillation where both interfaces move out of phase. These results explain the centering of liquid shells observed in several experiments.
    Keywords: FLUID MECHANICS AND HEAT TRANSFER
    Type: Physics of Fluids (ISSN 0031-9171); 30; 27-35
    Format: text
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