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Romain Brossier - One of the best experts on this subject based on the ideXlab platform.
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Feasibility study on 3D frequency-domain anisotropic elastic Wave Modeling using spectral element method with parallel direct linear solvers
2019Co-Authors: Romain Brossier, Ludovic MétivierAbstract:A feasibility study on 3D frequency-domain anisotropic elastic Wave Modeling is conducted. The spectral element method is applied to discretize the 3D frequency-domain anisotropic elastic Wave equation and the linear system is solved by parallel direct solvers, MUMPS andWSMP. A hybrid implementation of MPI and OpenMP for MUMPS is shown to be more efficient in flops and memory cost during the factorization. The influence of complex topography on MUMPS performance is negligible. With available resources, the largest scale Modeling, 30 Wavelengths in each dimension, is achieved. Using the block lowrank feature ofMUMPSleads to computational gains compared with the full-rank version. Limitation of MUMPS scalability for large number of MPI processes prompts us to investigate the performance of an alternative linear solver,WSMP. Preliminary comparison on small scale Modelings shows a better scalability of WSMP while being more computational demanding.
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3D Frequency-Domain Elastic Wave Modeling Using Spectral Element Method with a Parallel Direct Linear Solver
2019Co-Authors: Romain Brossier, Ludovic MétivierAbstract:Complex topography, free surface boundary condition and inelastic properties of media should be well considered for onshore geophysical prospecting. Thus an appropriate and accurate forward Modeling engine is very important. Unlike the time-domain implementation of many seismic imaging techniques, the counterpart in the frequency domain is rarely studied, despite of having many advantages, for example, only limited number of frequencies is needed for the inversion process, and solving the multiple-source problem is quite cheap if a direct solver is used. In this study, the spectral element method is applied to discretize the 3D frequency-domain anisotropic elastic Wave Modeling and the parallel direct solver MUMPS is used to solve the linear system. The structure and building process of the impedance matrix is thoroughly explained. We validate the numerical results by comparing with analytical solutions. A hybrid implementation of MPI and OpenMP for MUMPS is shown to be more efficient in flops and memory cost during the factorization. The influence of complex topography on MUMPS performance is negligible. With the available resources, the largest scale Modeling, 30 Wavelength in each dimension, is achieved. Other direct solvers and different low-rank techniques will also be investigated to reduce the flops and memory cost.
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Optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic Wave Modeling
Journal of Computational Physics, 2016Co-Authors: Bo Han, Ludovic Métivier, Romain BrossierAbstract:We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic Wave Modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg–Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poisson's ratios. Moreover, only 3.7 grid-points per minimum shear Wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain Modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates Wavefields having smaller error under the same discretization setups. Profiles of the Wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.
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2D and 3D frequency-domain elastic Wave Modeling in complex media with a parallel iterative solver
Geophysics, 2015Co-Authors: Ludovic Métivier, Romain Brossier, Bo Han, Jean VirieuxAbstract:Full-Waveform inversion and reverse time migration rely on an efficient forward-Modeling approach. Current 3D large-scale frequency-domain implementations of these techniques mostly extract the desired frequency component from the time-domain Wavefields through discrete Fourier transform. However, instead of conducting the time-marching steps for each seismic source, in which the time step is limited by the stability condition, performing the Wave Modeling directly in the frequency domain using an iterative linear solver may reduce the entire computational complexity. For 2D and 3D frequency-domain elastic Wave Modeling, a parallel iterative solver based on a conjugate gradient acceleration of the symmetric Kaczmarz row-projection method, named the conjugate-gradient-accelerated component-averaged row projections (CARP-CG) method, shows interesting convergence properties. The parallelization is realized through row-block division and component averaging operations. Convergence is achieved systematically even when different physical factors such as the space-dependent Poisson’s ratio, free-surface condition, and seismic attenuation are incorporated in the Wave Modeling. We determined that the scalability of CARP-CG was satisfactory, especially for large-scale applications, using up to several hundred computational cores. We found a potential improvement in computational complexity compared to time-domain Modeling through numerical experiments. Finally, we achieved a convergence at 5 Hz in a 3D heterogeneous model, involving fast-slow-fast layers resembling Waveguide geometries, with up to several hundred million unknowns, in fewer than 10 h on fewer than 200 cores. All of these results make CARP-CG a potential candidate of the forward Modeling engine for seismic imaging on challenging models.
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CARP-CG: A robust parallel iterative solver for frequency-domain elastic Wave Modeling, application to the Marmousi2 model
2014Co-Authors: Y. Liu, Ludovic Métivier, Romain Brossier, Bo Han, Jean VirieuxAbstract:Solving the frequency-domain elastic Wave equations relies on an efficient linear solver for the large, sparse, indefinite and ill-conditioned linear system derived from the discretization of the elastic Wave equation. Direct solvers, which are mostly based on LU decomposition, are efficient for multiple right-hand sides problems, but the memory requirement is huge due to the fill-in effects. On the contrary, iterative solvers fully benefit from the sparsity of the system, but they require problem-specific preconditioners to ensure the convergence because of the ill-conditioning of the system. In this study, we investigate the performance of a robust iterative method named CARP-CG for frequency-domain elastic Wave Modeling. CARP-CG method turns the original system into a symmetric positive semi-definite system by Kaczmarz row-projections. Such a system can be efficiently solved by the conjugate gradient (CG) method. The row-projections can be seen as a purely algebraic preconditioning technique which is general and is easy to implement. The parallelization is straightforward through a row-block decomposition combined with a component-averaging method. We discretize the 2D frequency-domain elastic Wave equation through a 4th order finite difference scheme. Numerical experiments on the Marmousi2 model exhibit a good scalability of CARP-CG. Comparisons between CARP-CG and standard Krylov iterative solvers (GMRES and CGNR) further emphasize the robustness and the fast convergence of CARP-CG method.
Ludovic Métivier - One of the best experts on this subject based on the ideXlab platform.
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Feasibility study on 3D frequency-domain anisotropic elastic Wave Modeling using spectral element method with parallel direct linear solvers
2019Co-Authors: Romain Brossier, Ludovic MétivierAbstract:A feasibility study on 3D frequency-domain anisotropic elastic Wave Modeling is conducted. The spectral element method is applied to discretize the 3D frequency-domain anisotropic elastic Wave equation and the linear system is solved by parallel direct solvers, MUMPS andWSMP. A hybrid implementation of MPI and OpenMP for MUMPS is shown to be more efficient in flops and memory cost during the factorization. The influence of complex topography on MUMPS performance is negligible. With available resources, the largest scale Modeling, 30 Wavelengths in each dimension, is achieved. Using the block lowrank feature ofMUMPSleads to computational gains compared with the full-rank version. Limitation of MUMPS scalability for large number of MPI processes prompts us to investigate the performance of an alternative linear solver,WSMP. Preliminary comparison on small scale Modelings shows a better scalability of WSMP while being more computational demanding.
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3D Frequency-Domain Elastic Wave Modeling Using Spectral Element Method with a Parallel Direct Linear Solver
2019Co-Authors: Romain Brossier, Ludovic MétivierAbstract:Complex topography, free surface boundary condition and inelastic properties of media should be well considered for onshore geophysical prospecting. Thus an appropriate and accurate forward Modeling engine is very important. Unlike the time-domain implementation of many seismic imaging techniques, the counterpart in the frequency domain is rarely studied, despite of having many advantages, for example, only limited number of frequencies is needed for the inversion process, and solving the multiple-source problem is quite cheap if a direct solver is used. In this study, the spectral element method is applied to discretize the 3D frequency-domain anisotropic elastic Wave Modeling and the parallel direct solver MUMPS is used to solve the linear system. The structure and building process of the impedance matrix is thoroughly explained. We validate the numerical results by comparing with analytical solutions. A hybrid implementation of MPI and OpenMP for MUMPS is shown to be more efficient in flops and memory cost during the factorization. The influence of complex topography on MUMPS performance is negligible. With the available resources, the largest scale Modeling, 30 Wavelength in each dimension, is achieved. Other direct solvers and different low-rank techniques will also be investigated to reduce the flops and memory cost.
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Optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic Wave Modeling
Journal of Computational Physics, 2016Co-Authors: Bo Han, Ludovic Métivier, Romain BrossierAbstract:We investigate an optimal fourth-order staggered-grid finite-difference scheme for 3D frequency-domain viscoelastic Wave Modeling. An anti-lumped mass strategy is incorporated to minimize the numerical dispersion. The optimal finite-difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity. An iterative damped least-squares method, the Levenberg–Marquardt algorithm, is utilized for the optimization. Dispersion analysis shows that the optimal fourth-order scheme presents less grid dispersion and anisotropy than the conventional fourth-order scheme with respect to different Poisson's ratios. Moreover, only 3.7 grid-points per minimum shear Wavelength are required to keep the error of the group velocities below 1%. The memory cost is then greatly reduced due to a coarser sampling. A parallel iterative method named CARP-CG is used to solve the large ill-conditioned linear system for the frequency-domain Modeling. Validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions. Compared with the conventional fourth-order scheme, the optimal scheme generates Wavefields having smaller error under the same discretization setups. Profiles of the Wavefields are presented to confirm better agreement between the optimal results and the analytic solutions.
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2D and 3D frequency-domain elastic Wave Modeling in complex media with a parallel iterative solver
Geophysics, 2015Co-Authors: Ludovic Métivier, Romain Brossier, Bo Han, Jean VirieuxAbstract:Full-Waveform inversion and reverse time migration rely on an efficient forward-Modeling approach. Current 3D large-scale frequency-domain implementations of these techniques mostly extract the desired frequency component from the time-domain Wavefields through discrete Fourier transform. However, instead of conducting the time-marching steps for each seismic source, in which the time step is limited by the stability condition, performing the Wave Modeling directly in the frequency domain using an iterative linear solver may reduce the entire computational complexity. For 2D and 3D frequency-domain elastic Wave Modeling, a parallel iterative solver based on a conjugate gradient acceleration of the symmetric Kaczmarz row-projection method, named the conjugate-gradient-accelerated component-averaged row projections (CARP-CG) method, shows interesting convergence properties. The parallelization is realized through row-block division and component averaging operations. Convergence is achieved systematically even when different physical factors such as the space-dependent Poisson’s ratio, free-surface condition, and seismic attenuation are incorporated in the Wave Modeling. We determined that the scalability of CARP-CG was satisfactory, especially for large-scale applications, using up to several hundred computational cores. We found a potential improvement in computational complexity compared to time-domain Modeling through numerical experiments. Finally, we achieved a convergence at 5 Hz in a 3D heterogeneous model, involving fast-slow-fast layers resembling Waveguide geometries, with up to several hundred million unknowns, in fewer than 10 h on fewer than 200 cores. All of these results make CARP-CG a potential candidate of the forward Modeling engine for seismic imaging on challenging models.
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CARP-CG: A robust parallel iterative solver for frequency-domain elastic Wave Modeling, application to the Marmousi2 model
2014Co-Authors: Y. Liu, Ludovic Métivier, Romain Brossier, Bo Han, Jean VirieuxAbstract:Solving the frequency-domain elastic Wave equations relies on an efficient linear solver for the large, sparse, indefinite and ill-conditioned linear system derived from the discretization of the elastic Wave equation. Direct solvers, which are mostly based on LU decomposition, are efficient for multiple right-hand sides problems, but the memory requirement is huge due to the fill-in effects. On the contrary, iterative solvers fully benefit from the sparsity of the system, but they require problem-specific preconditioners to ensure the convergence because of the ill-conditioning of the system. In this study, we investigate the performance of a robust iterative method named CARP-CG for frequency-domain elastic Wave Modeling. CARP-CG method turns the original system into a symmetric positive semi-definite system by Kaczmarz row-projections. Such a system can be efficiently solved by the conjugate gradient (CG) method. The row-projections can be seen as a purely algebraic preconditioning technique which is general and is easy to implement. The parallelization is straightforward through a row-block decomposition combined with a component-averaging method. We discretize the 2D frequency-domain elastic Wave equation through a 4th order finite difference scheme. Numerical experiments on the Marmousi2 model exhibit a good scalability of CARP-CG. Comparisons between CARP-CG and standard Krylov iterative solvers (GMRES and CGNR) further emphasize the robustness and the fast convergence of CARP-CG method.
Prashant Kumar - One of the best experts on this subject based on the ideXlab platform.
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Spectral Wave Modeling of tsunami Waves in Pohang New Harbor (South Korea) and Paradip Port (India)
Ocean Dynamics, 2020Co-Authors: Prashant KumarAbstract:A coupled numerical model is developed based on the spectral element method (SEM) and boundary element method (BEM) to predict the characteristics of tsunami Wave response on Pohang New Harbor (PNH), South Korea, and Paradip Port, India. The current numerical model is developed to analyze the impacts on the coastal region with slowly varying bathymetry under the resonance conditions. In this method, the boundary integral equation of each boundary segment is discretized into the spectral elements using the Chebyshev’s polynomial, and the corresponding boundary integrals are transformed using the Jacobian of transformation. This leads to a highly accurate depiction of the boundary edges or corners of the complex domain. Convergence and error analyses are conducted on the rectangular port using the BEM and the present method, which shows that the later approach enhances the overall numerical accuracy of the model. The simulation results for normal day Wave (ordinary Wave propagating in the ocean under normal conditions) and tsunami Waves are validated with previous studies and measurement data at Pohang New Harbor (PNH), South Korea. In addition, the calculated spectral density for tsunami Waves at PNH is also compared with the Hokkaido tsunami (reported on July 12, 1993) data at tide gauge record station in Pohang. The amplification factor is determined for normal day Wave and tsunami Waves at key locations inside the PNH and Paradip Port for different directional incident Waves. Further, the spectral density is also estimated at the same locations with respect to the Wave period to understand the consequences of tsunami Waves on the coastal ports. Therefore, the coupled numerical model is an efficient tool to predict tsunami Wave impact on a realistic port. The causes and countermeasure are suggested to reduce the risk of normal day Waves and tsunami Waves.
Stéphane Operto - One of the best experts on this subject based on the ideXlab platform.
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A Robust Absorbing Layer Method for Seismic Wave Simulation in Anisotropic Media
2014Co-Authors: Ludovic Métivier, Stéphane Operto, Romain Brossier, Jean VirieuxAbstract:Seismic Wave Modeling requires using adapted boundary conditions to simulate infinite or semi-infinite media. Because of its efficiency, the Perfectly Matched Layers (PML) method has rapidly become the standard for acoustic and elastic propagation. However, PML are not adapted to anisotropic media for which the method becomes amplifying. Alternative methods have to be designed. In this study, we present the SMART layer method, which relies on a diagonal decomposition of the hyperbolic operator. The method is not perfectly matched, therefore less efficient than the PML method, however it is proved to remain dissipative, even for anisotropic media. We apply the method to the acoustic TTI equations. We present numerical results on a homogeneous test case and on the BP 2007 model, which includes a space dependent tilt angle. We compare the SMART and the PML methods. The results emphasize the robustness of the SMART method: no Wave amplification is observed. In addition, the accuracy of the PML can be reached at the expense of an increase of the SMART layer width. The additional computational cost is compensated by the simple form of the SMART layer: only a zero-order term is added to the equations and no additional variables are required
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Frequency-domain Acoustic Wave Modeling Using a Hybrid Direct-iterative Solver Based on a Domain Decomposition Method
70th EAGE Conference and Exhibition incorporating SPE EUROPEC 2008, 2014Co-Authors: F Sourbier, Abdulhamid Haidar, Stéphane Operto, Luc Giraud, Jean VirieuxAbstract:Designing an efficient Modeling tool is a key point for large 3D frequency-domain full-Waveform inversion problems. We present a frequency-domain acoustic Wave Modeling using a hybrid direct-iterative solver based on a parallel domain decomposition method and Schur complement approach. The main interest of mixing solvers is to overcome the huge memory complexity of direct solvers while partially preserving efficient multi-RHS simulations and mitigating the iteration count in iterative solvers. To improve the convergence rate of the iterative solver, a preconditioning based on an additive Schwartz approach is used. Discretization of the Helmholtz equation is based on a parsimonious finite-difference method but the domain decomposition method could apply to any numerical scheme such as finite-element or finite-volume methods and to any media such as elastic, anisotropic ones ... To asses the efficiency of the hybrid approach, we computed simulations in the 2D Marmousi II and 3D SEG/EAGE Overthrust model, and compared results with that of a direct solver.
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seismic Wave Modeling for seismic imaging
Geophysics, 2009Co-Authors: Jean Virieux, V Etienne, F Sourbier, Romain Brossier, Stéphane Operto, Luc Giraud, Azzam HaidarAbstract:Many scientific applications require accurate Modeling of seismic Wave propagation in complex media. These objectives can include fundamental understanding of seismic Wave propagation of the Earth on a global scale, including fluid envelopes, mitigation of seismic risk with better quantitative estimates of seismic hazard, and improved exploitation of the natural resources in the crust of the Earth. Accurate quantification is a continual quest, and benchmark protocols have been designed for model definitions and comparison of solutions. Through this quest, an impressive number of numerical tools have been developed, ranging from efficient finite-difference methods to more sophisticated finite-element methods, and including the so-called pseudospectral methods (see Wu and Maupin for a review). The main motivation behind these permanent developments has been to improve the efficiency and accuracy of forward Modeling. To achieve this, one systematic choice for both finite-difference and finite-element methods has involved explicit time-stepping integration to avoid matrix inversion...
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Parsimonious finite-volume frequency-domain method for 2D P-SV-Wave Modeling
Geophysical Journal International, 2008Co-Authors: Romain Brossier, Jean Virieux, Stéphane OpertoAbstract:A new numerical technique for solving the 2D elastodynamic equations based on a finite volume approach is proposed. The associated discretization is through triangles. Only fluxes of required quantities are shared between cells, relaxing meshing conditions compared to finite element methods. The free surface is described along the edges of the triangles which may have different slopes. By applying a parsimonious strategy, stress components are eliminated from the discrete equations and only velocities are left as unknowns in triangles, minimizing the core memory requirement of the simulation. Efficient PML absorbing conditions have been designed for damping Waves around the grid. Since the technique is devoted to full Waveform inversion, we implemented the method in the frequency domain using a direct solver, an efficient strategy for multiple-source simulations. Standard dispersion analysis in infinite homogeneous media shows that numerical dispersion is similar to those of O(¢x2) staggeredgrid finite-difference formulations when considering structured triangular meshes. The method is validated against analytical solutions of several canonical problems and with numerical solutions computed with a well-established finite-difference time-domain method in heterogeneous media. In presence of a free surface, the finite-volume method requires ten triangles per Wavelength for a flat topography and fifteen triangles per Wavelength for more complex shapes, well below criteria required by the staircase approximation of finite-difference methods. Comparison between the frequency-domain finite-volume and the O(¢x2) rotated finite-difference methods also shows that the former is faster and less-memory demanding for a given accuracy level. We developed an efficient method for 2-D P-SV-Wave Modeling on structured triangular meshes as a tool for frequency-domain full-Waveform inversion. Further work is required to assess the method on unstructured meshes.
Dan Jiao - One of the best experts on this subject based on the ideXlab platform.
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an h 2 matrix based integral equation solver of linear complexity for large scale full Wave Modeling of 3d circuits
Electrical Performance of Electronic Packaging, 2008Co-Authors: Wenwen Chai, Dan JiaoAbstract:State-of-the-art integral-equation based computational electromagnetic methods rely on techniques that can perform a matrix-vector multiplication in O(N log N) operations, with N being the matrix size. In this work, a fast integral-equation-based solver was developed for full-Wave Modeling of large-scale complicated 3D circuits. Both memory consumption and time complexity were shown to be O(N). The superior performance applies to any arbitrarily shaped 3D structure. Numerical results demonstrate its accuracy and efficiency.
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a unified finite element solution from zero frequency to microWave frequencies for full Wave Modeling of large scale three dimensional on chip interconnect structures
IEEE Antennas and Propagation Society International Symposium, 2008Co-Authors: Dan JiaoAbstract:It has been observed that a full-Wave finite-element-based solution breaks down at low frequencies. This hinders its application to on-chip problems in which broadband Modeling from DC to microWave frequencies is required. Although a static formulation and a full-Wave formulation can be stitched together to solve this problem, it is cumbersome to implement both static and full-Wave solvers and make transitions between these two when necessary. In this work, a unified finite-element solution from zero frequency to microWave frequencies is developed for full-Wave Modeling of large-scale three-dimensional on-chip interconnect structures. In this solution, a single full-Wave formulation is used. No switching to a static formulation is needed at low frequencies. Numerical and experimental results demonstrate its validity.
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a unified finite element solution from zero frequency to microWave frequencies for full Wave Modeling of large scale three dimensional on chip interconnect structures
IEEE Transactions on Advanced Packaging, 2008Co-Authors: Dan JiaoAbstract:It has been observed that a full-Wave finite-element-based solution breaks down at low frequencies. This hinders its application to on-chip problems in which broadband Modeling from direct current to microWave frequencies is required. Although a static formulation and a full-Wave formulation can be stitched together to solve this problem, it is cumbersome to implement both static and full-Wave solvers and make transitions between these two when necessary. In this work, a unified finite-element solution from zero frequency to microWave frequencies is developed for full-Wave Modeling of large-scale three-dimensional on-chip interconnect structures. In this solution, a single full-Wave formulation is used. No switching to a static formulation is needed at low frequencies. This is achieved by first identifying the reason why a full-Wave eigenvalue-based solution breaks down at low frequencies, and then developing an approach to eliminate the reason. The low frequency breakdown problem was found to be attributed to the discrepant frequency dependence of the real part and the imaginary part of the eigenvalues, which leads to an ill-conditioned eigenvalue system at low frequencies. The discrepant frequency dependence of the real part and the imaginary part is further attributed to the different scaling of the transverse and longitudinal fields with respect to frequency in a transmission-line type structure. By extracting transverse and longitudinal fields separately in the framework of a full-Wave formulation, we avoid the numerical difficulty of solving an ill-conditioned eigen-system at low frequencies. The validity of the proposed approach is demonstrated by numerical and experimental results.
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a novel technique for full Wave Modeling of large scale three dimensional high speed on off chip interconnect structures
International Conference on Simulation of Semiconductor Processes and Devices, 2003Co-Authors: Dan Jiao, M Mazumder, S Chakravarty, Changhong Dai, Mauro J Kobrinsky, M Harmes, Scott ListAbstract:This paper presents a novel, rigorous, and fast method for full-Wave Modeling of high-speed interconnect structures. In this method, the original Wave propagation problem is represented into a generalized eigenvalue problem. The resulting eigenvalue representation can comprehend conductor and dielectric losses, arbitrary dielectric and conductor configurations, and arbitrary materials such as dispersive, and anisotropic media. The edge basis function is employed to accurately represent the unknown field, and the triangular element is adopted to flexibly model arbitrary geometry. A mode-matching technique applicable to lossy system is developed to solve large-scale 3D problems by using 2D-like CPU time and memory. A circuit-based extraction technique is developed to obtain S-parameters from the unknown fields. The proposed technique can generate S-parameters, full-Wave RLGC, propagation constants, characteristic impedances, voltage, current, and field distributions, and hence yield a comprehensive representation of interconnect structures. Experimental and numerical results demonstrate its accuracy and efficiency.
