Transactions of the Institute of electronics, information and communication engineers, 1992
One of the major problems in simulating stress-dependent oxidation is slow convergence behavior o... more One of the major problems in simulating stress-dependent oxidation is slow convergence behavior or nonconvergence because of high nonlinearities, especially in the first few time steps, even in a simple initial structure for LOCOS. In this paper we present useful strategies to get stable convergence in a fully stress-dependent finite element oxidation simulator like SUPREM-IV [1]. The s t ra tegies are Firs t , in the stress-dependent oxidation simulation, we have to solve the coupled nonlinear equations of incompressible oxide flow and oxidant diffusion. There are two standard ways to treat the incompressible flow problciri [2]. One is the mixed formulation (unknown variables are velocity and pressure), and the other is the penalty function formulation (unknown variable is velocity only). SUPREM-IV (version 8912) uses the second approach. By using a penalty function method, pressure variable is eliminated from the system of equations which reduces the computational cost and elimina...
We consider the performance of the finite element method on a vector supercomputer. The computati... more We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP.
There are several significant differences that the user familiar with past versions of ConMan wil... more There are several significant differences that the user familiar with past versions of ConMan will find in the 3.0 version. First, we removed the clunky memory manager library (a set of routines wrapped around the c function malloc) and replaced them with FORTRAN 90's allocate and deallocate functions. This eliminates many of the compilation problems people experienced with the 2.0 version. Most of these routines were in files subroutines input and elminp. As part of a general clean up, we replaced the separate input and elminp (both input subroutines) and created a new input subroutine. As part of this we removed the `element library' function (eglib.F and eg2.F) which was a structure originally designed for different formulations 2D Cartesian, 3D Cartesian, spherical axysymmetric, ...). Because these were never fully developed, it made on sense to retain the cumbersome structure. We also moved all out the subroutines associated with output, into new output subroutine. The ...
By use of "conjugate gradients" technique, concurrent efficiency greater than 90 percen... more By use of "conjugate gradients" technique, concurrent efficiency greater than 90 percent. Algorithm applies method of conjugate gradients to iterative solution of finite-element problems on concurrent processor. With algorithm, iteration rates nearly proportional to number of processors. For sufficiently large problems, fraction of proportional speedup achieved, called concurrent efficiency, exceeds 90 percent. Results indicate future application of this and related algorithms to large finite-element problems depend primarily upon applicability of iteratiave techniques, not upon issues of concurrency or efficiency.
We review experiences gained in the development of a commercial, engineering analysis software sy... more We review experiences gained in the development of a commercial, engineering analysis software system for parallel computation. The lessons learned frequently complement, but occasionally are in conflict with academic ideas about parallel computations. The thoughts presented are believed to be useful to those engaged in, or about to embark upon, parallel implementations aimed at the industrial sector. 0 1997 Elsevier Science B.V.
We explain the global variation of Benioff zone seismicity with depth and the orientation of stre... more We explain the global variation of Benioff zone seismicity with depth and the orientation of stress axes of deep and intermediate earthquakes using numerical models of subducting slabs. Models that match the seismicity and stress require a barrier to flow at the 670 km seismic discontinuity. The barrier may be a viscosity increase of at least an order of magnitude or a chemical discontinuity. Instantaneous flow is subparallel to the slabs for models with a viscosity increase but contorted for models with a chemical barrier. Log N (number of earthquakes) decreases linearly to 250-300 km depth and increases thereafter. Stress magnitude in our models shows the same pattern, in accord with experiments showing N proportional to e k~, with k a constant and a stress magnitude. The models predict downdip compression in the slabs at depths below 300-400 km, as observed for earthquake stress axes.
Abstract This paper describes an implementation of a generalized profile/sparse solution method f... more Abstract This paper describes an implementation of a generalized profile/sparse solution method for finite element analysis. The solution procedure can be used as a profile solver as well as a sparse matrix solver, depending on the physical model and the ordering scheme used to number the system of equations. The data structure employed in the implementation is discussed in detail. The procedures for element stiffness assembly, symbolic and numeric factorization of the global stiffness matrix and the calculation of the displacement solution vector are described. Experimental results are presented to demonstrate the effectiveness of this profile/sparse solution scheme.
This paper demonstrates the utility, generality and simplicity of a computational method of solvi... more This paper demonstrates the utility, generality and simplicity of a computational method of solving systems of ordinary differential equations. The central idea of the method revolves around the ability to generate a numerical approximation of the general solution of systems of linear differential equations. The idea of obtaining a numerical approximation to the general solution leads to changes in the traditional approaches for solving boundary value problems. The method is extended to nonlinear boundary value problems and to boundary value problems with a boundary condition given at infinity. The estimation of unknown parameters in a dynamical system is also treated.
Computer Methods in Applied Mechanics and Engineering, 1993
A new formulation employing the Galerkin/least-squares finite element method is presented for the... more A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional onecarrier n ÷-n.n + devices, The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.
Computer Methods in Applied Mechanics and Engineering, 1995
A space-time Galerkin/least-squares finite element method was presented in [l] for numerical simu... more A space-time Galerkin/least-squares finite element method was presented in [l] for numerical simulation of single-carrier hydrodynamic semiconductor device equations. The single-carrier hydrodynamic device equations were shown to resemble the ideal gas equations and Galerkin/least-squares finite element method, originally developed for computational fluid dynamics equations [16], was extended to solve semiconductor device applications. In this paper, the space-time Galerkin/least-squares finite element method is further extended and generalized to solve two-carrier hydrodynamic device equations. The proposed formulation is based on a time-discontinuous Galerkin method, in which physical entropy variables are employed. A standard Galerkin finite element method is applied to the Poisson equation. Numerical simulations are performed on the coupled Poisson and the two-carrier hydrodynamic equations employing a staggered approach. A mathematical analysis of the time-dependent multi-dimensional hydrodynamic model is performed to determine well-posed boundary conditions for electrical contacts. The number of boundary conditions that need to be specified for the hydrodynamic equations at inflow and outflow boundaries of the device are derived. Example boundary conditions that are based either on physical and/or mathematical basis are presented. Stability of the numerical algorithms is addressed. The space-time Galerkin/least-squares finite element method and the standard Galerkin finite element method for the hydrodynamic and Poisson equations, respectively, are shown to be stable. Specifically, a Clausius-Duhem inequality, a basic stability requirement, is derived for the hydrodynamic equations and the proposed numerical method automatically satisfies this stability requirement. Numerical simulations are performed on one-and two-dimensional two-carrier p-n diodes and the results demonstrate the effectiveness of the proposed numerical method.
ABSTRACT Large CFD models require memory sizes larger than can be supported by today’s single ‘no... more ABSTRACT Large CFD models require memory sizes larger than can be supported by today’s single ‘node’ computers. Using the memory of more than one node can greatly complicate the creation of a well-performing program. We believe that preserving globally addressable memory beyond the boundary of a single node enables diversity of programming methods and provides flexibility essential for optimum algorithm development. We isolate the effects of algorithm and implementation by porting the same parallel CFD algorithm in three styles: fine-grain shared memory, coarse-grain shared memory, and coarse-grain distributed memory. Despite some fundamental differences in programming implementation, both distributed memory and coarse-grain shared memory code provide very close parallel performance due to algorithmic similarities. At the same time fine-grain shared memory code, despite use of the same programming paradigm as coarse-grain shared memory program, falls far behind due to unavoidable parallel performance penalties caused by Amdahl’s law and some other limitations.
Physics of the Earth and Planetary Interiors, 1990
ConMan: vectorizing a finite element code for incompressible two-dimensional convection in the Ea... more ConMan: vectorizing a finite element code for incompressible two-dimensional convection in the Earth's mantle. Phys. Earth Planet. Inter., We discuss some simple concepts for vectorizing scientific codes, then apply these concepts to ConMan, a finite element code for simulations of mantle convection. We demonstrate that large speed-ups, close to the theoretical limit of the machine, are possible for entire codes, not just specially constructed routines. Although our specific code uses the finite element method, the vectorizing concepts discussed are widely applicable. Present address: Department of Earth. Atmosphenc and Planetary Sciences, Massachusetts Institute of Technology, chaining and unrolling. Although we illustrate Cambridge, MA 02139, U.S.A. vectorization with the finite element code Con-0031-9201/90/$03.50
Bulletin of the Seismological Society of America, Dec 1, 1984
Three-dimensional finite element modeling techniques are used to synthesize geodetic and seismolo... more Three-dimensional finite element modeling techniques are used to synthesize geodetic and seismological results for 1979 Imperial Valley earthquake. The strategy pursued consists of two principal steps. In the first step, the seismologically-derived coseismic fault slip is taken as a function of position in the fault plane and is applied directly to the three-dimensional dislocation model. In the second step, a physical model of stresses and constitutive parameters is perturbed so as to reproduce the observed fault slip. Hence, the principal features of the coseismic slip pattern are explained by a stress-driven fault model in which: (1) a spatially unresolved asperity is found equivalent to a stress drop of 18 MPa averaged over an area of 15 sq km, and (2) driving stress is essentially absent on the fault segment overlapping the 1940 earthquake rupture zone.
... nia's Departmunt of Commerce uwder grant number C90-072 I oo For and by DARPA. 2 Ref... more ... nia's Departmunt of Commerce uwder grant number C90-072 I oo For and by DARPA. 2 References ".. .. ii] [11] MR Pinto, CS Rafferty, HA Yeager. and RW Dutton, . "PISCES-JIB Supplementary Rqxpr" Stanford Electronics ----. Lab., Stanford Univ., Stanford, CA, 1985. ...
Transactions of the Institute of electronics, information and communication engineers, 1992
One of the major problems in simulating stress-dependent oxidation is slow convergence behavior o... more One of the major problems in simulating stress-dependent oxidation is slow convergence behavior or nonconvergence because of high nonlinearities, especially in the first few time steps, even in a simple initial structure for LOCOS. In this paper we present useful strategies to get stable convergence in a fully stress-dependent finite element oxidation simulator like SUPREM-IV [1]. The s t ra tegies are Firs t , in the stress-dependent oxidation simulation, we have to solve the coupled nonlinear equations of incompressible oxide flow and oxidant diffusion. There are two standard ways to treat the incompressible flow problciri [2]. One is the mixed formulation (unknown variables are velocity and pressure), and the other is the penalty function formulation (unknown variable is velocity only). SUPREM-IV (version 8912) uses the second approach. By using a penalty function method, pressure variable is eliminated from the system of equations which reduces the computational cost and elimina...
We consider the performance of the finite element method on a vector supercomputer. The computati... more We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP.
There are several significant differences that the user familiar with past versions of ConMan wil... more There are several significant differences that the user familiar with past versions of ConMan will find in the 3.0 version. First, we removed the clunky memory manager library (a set of routines wrapped around the c function malloc) and replaced them with FORTRAN 90's allocate and deallocate functions. This eliminates many of the compilation problems people experienced with the 2.0 version. Most of these routines were in files subroutines input and elminp. As part of a general clean up, we replaced the separate input and elminp (both input subroutines) and created a new input subroutine. As part of this we removed the `element library' function (eglib.F and eg2.F) which was a structure originally designed for different formulations 2D Cartesian, 3D Cartesian, spherical axysymmetric, ...). Because these were never fully developed, it made on sense to retain the cumbersome structure. We also moved all out the subroutines associated with output, into new output subroutine. The ...
By use of "conjugate gradients" technique, concurrent efficiency greater than 90 percen... more By use of "conjugate gradients" technique, concurrent efficiency greater than 90 percent. Algorithm applies method of conjugate gradients to iterative solution of finite-element problems on concurrent processor. With algorithm, iteration rates nearly proportional to number of processors. For sufficiently large problems, fraction of proportional speedup achieved, called concurrent efficiency, exceeds 90 percent. Results indicate future application of this and related algorithms to large finite-element problems depend primarily upon applicability of iteratiave techniques, not upon issues of concurrency or efficiency.
We review experiences gained in the development of a commercial, engineering analysis software sy... more We review experiences gained in the development of a commercial, engineering analysis software system for parallel computation. The lessons learned frequently complement, but occasionally are in conflict with academic ideas about parallel computations. The thoughts presented are believed to be useful to those engaged in, or about to embark upon, parallel implementations aimed at the industrial sector. 0 1997 Elsevier Science B.V.
We explain the global variation of Benioff zone seismicity with depth and the orientation of stre... more We explain the global variation of Benioff zone seismicity with depth and the orientation of stress axes of deep and intermediate earthquakes using numerical models of subducting slabs. Models that match the seismicity and stress require a barrier to flow at the 670 km seismic discontinuity. The barrier may be a viscosity increase of at least an order of magnitude or a chemical discontinuity. Instantaneous flow is subparallel to the slabs for models with a viscosity increase but contorted for models with a chemical barrier. Log N (number of earthquakes) decreases linearly to 250-300 km depth and increases thereafter. Stress magnitude in our models shows the same pattern, in accord with experiments showing N proportional to e k~, with k a constant and a stress magnitude. The models predict downdip compression in the slabs at depths below 300-400 km, as observed for earthquake stress axes.
Abstract This paper describes an implementation of a generalized profile/sparse solution method f... more Abstract This paper describes an implementation of a generalized profile/sparse solution method for finite element analysis. The solution procedure can be used as a profile solver as well as a sparse matrix solver, depending on the physical model and the ordering scheme used to number the system of equations. The data structure employed in the implementation is discussed in detail. The procedures for element stiffness assembly, symbolic and numeric factorization of the global stiffness matrix and the calculation of the displacement solution vector are described. Experimental results are presented to demonstrate the effectiveness of this profile/sparse solution scheme.
This paper demonstrates the utility, generality and simplicity of a computational method of solvi... more This paper demonstrates the utility, generality and simplicity of a computational method of solving systems of ordinary differential equations. The central idea of the method revolves around the ability to generate a numerical approximation of the general solution of systems of linear differential equations. The idea of obtaining a numerical approximation to the general solution leads to changes in the traditional approaches for solving boundary value problems. The method is extended to nonlinear boundary value problems and to boundary value problems with a boundary condition given at infinity. The estimation of unknown parameters in a dynamical system is also treated.
Computer Methods in Applied Mechanics and Engineering, 1993
A new formulation employing the Galerkin/least-squares finite element method is presented for the... more A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional onecarrier n ÷-n.n + devices, The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.
Computer Methods in Applied Mechanics and Engineering, 1995
A space-time Galerkin/least-squares finite element method was presented in [l] for numerical simu... more A space-time Galerkin/least-squares finite element method was presented in [l] for numerical simulation of single-carrier hydrodynamic semiconductor device equations. The single-carrier hydrodynamic device equations were shown to resemble the ideal gas equations and Galerkin/least-squares finite element method, originally developed for computational fluid dynamics equations [16], was extended to solve semiconductor device applications. In this paper, the space-time Galerkin/least-squares finite element method is further extended and generalized to solve two-carrier hydrodynamic device equations. The proposed formulation is based on a time-discontinuous Galerkin method, in which physical entropy variables are employed. A standard Galerkin finite element method is applied to the Poisson equation. Numerical simulations are performed on the coupled Poisson and the two-carrier hydrodynamic equations employing a staggered approach. A mathematical analysis of the time-dependent multi-dimensional hydrodynamic model is performed to determine well-posed boundary conditions for electrical contacts. The number of boundary conditions that need to be specified for the hydrodynamic equations at inflow and outflow boundaries of the device are derived. Example boundary conditions that are based either on physical and/or mathematical basis are presented. Stability of the numerical algorithms is addressed. The space-time Galerkin/least-squares finite element method and the standard Galerkin finite element method for the hydrodynamic and Poisson equations, respectively, are shown to be stable. Specifically, a Clausius-Duhem inequality, a basic stability requirement, is derived for the hydrodynamic equations and the proposed numerical method automatically satisfies this stability requirement. Numerical simulations are performed on one-and two-dimensional two-carrier p-n diodes and the results demonstrate the effectiveness of the proposed numerical method.
ABSTRACT Large CFD models require memory sizes larger than can be supported by today’s single ‘no... more ABSTRACT Large CFD models require memory sizes larger than can be supported by today’s single ‘node’ computers. Using the memory of more than one node can greatly complicate the creation of a well-performing program. We believe that preserving globally addressable memory beyond the boundary of a single node enables diversity of programming methods and provides flexibility essential for optimum algorithm development. We isolate the effects of algorithm and implementation by porting the same parallel CFD algorithm in three styles: fine-grain shared memory, coarse-grain shared memory, and coarse-grain distributed memory. Despite some fundamental differences in programming implementation, both distributed memory and coarse-grain shared memory code provide very close parallel performance due to algorithmic similarities. At the same time fine-grain shared memory code, despite use of the same programming paradigm as coarse-grain shared memory program, falls far behind due to unavoidable parallel performance penalties caused by Amdahl’s law and some other limitations.
Physics of the Earth and Planetary Interiors, 1990
ConMan: vectorizing a finite element code for incompressible two-dimensional convection in the Ea... more ConMan: vectorizing a finite element code for incompressible two-dimensional convection in the Earth's mantle. Phys. Earth Planet. Inter., We discuss some simple concepts for vectorizing scientific codes, then apply these concepts to ConMan, a finite element code for simulations of mantle convection. We demonstrate that large speed-ups, close to the theoretical limit of the machine, are possible for entire codes, not just specially constructed routines. Although our specific code uses the finite element method, the vectorizing concepts discussed are widely applicable. Present address: Department of Earth. Atmosphenc and Planetary Sciences, Massachusetts Institute of Technology, chaining and unrolling. Although we illustrate Cambridge, MA 02139, U.S.A. vectorization with the finite element code Con-0031-9201/90/$03.50
Bulletin of the Seismological Society of America, Dec 1, 1984
Three-dimensional finite element modeling techniques are used to synthesize geodetic and seismolo... more Three-dimensional finite element modeling techniques are used to synthesize geodetic and seismological results for 1979 Imperial Valley earthquake. The strategy pursued consists of two principal steps. In the first step, the seismologically-derived coseismic fault slip is taken as a function of position in the fault plane and is applied directly to the three-dimensional dislocation model. In the second step, a physical model of stresses and constitutive parameters is perturbed so as to reproduce the observed fault slip. Hence, the principal features of the coseismic slip pattern are explained by a stress-driven fault model in which: (1) a spatially unresolved asperity is found equivalent to a stress drop of 18 MPa averaged over an area of 15 sq km, and (2) driving stress is essentially absent on the fault segment overlapping the 1940 earthquake rupture zone.
... nia's Departmunt of Commerce uwder grant number C90-072 I oo For and by DARPA. 2 Ref... more ... nia's Departmunt of Commerce uwder grant number C90-072 I oo For and by DARPA. 2 References ".. .. ii] [11] MR Pinto, CS Rafferty, HA Yeager. and RW Dutton, . "PISCES-JIB Supplementary Rqxpr" Stanford Electronics ----. Lab., Stanford Univ., Stanford, CA, 1985. ...
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Papers by Arthur Raefsky