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Particle-Based Device Simulation Methods
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
In FMM, multipole moments are used to represent distant particle groups and a local expansion is used to evaluate the contribution from distant particles in the form of a series. The multipole moment associated with a distant group can be translated into the coefficient of the local expansion associated with a local group. In FMM, the computational domain is decomposed in a hierarchical manner with a quad-tree in two dimensions and an oct-tree in three dimensions to carry out efficient and systematic grouping of particles with tree structures. The hierarchical decomposition is used to cluster particles at various spatial lengths and compute interactions with other clusters that are sufficiently far away by means of the series expansions.
High-Accuracy Parasitic Extraction
Published in Louis Scheffer, Luciano Lavagno, Grant Martin, EDA for IC Implementation, Circuit Design, and Process Technology, 2018
FMM methods reduce the computation time and memory by orders of magnitude for even modest capacitance problems. It was shown in [6] that even a small problem of roughly 6000 panels could be solved 10 times faster than a dense matrix–vector product iterative scheme, and 100 times faster than direct factorization.
A simple iterative geometry-based interface-preserving reinitialization for the level set method
Published in International Journal of Computational Fluid Dynamics, 2019
Lanhao Zhao, Hairong Zhang, Jia Mao, Hanyue Zhu, Dawei Peng, Kailong Mu
Reinitialization techniques can be divided into explicit and implicit schemes. The explicit reinitialization requires the location of zero-level-set before reconstructing a SDF. The first attempt to reinitialize the level set function explicitly is the direct approach (Chopp 1993; Shakoor et al. 2015; Long and Choi 2017) introduced by Chopp. The direct method calculates the signed distance from each node to the interface but tends to produce irregular distribution of the level set especially at sharp corners. Hence, the direct method cannot provide a smooth description of the sharp interface. Another explicit scheme is the fast marching method (FMM) (Sethian 1996; Bockmann and Vartdal 2014; Yang and Stern 2017) proposed by Sethian. This method generates a SDF via the eikonal equation based on the appropriate construction of extension velocities, but the exact location of the interface must be represented explicitly to construct the expansion velocities in each pseudo time step. Moreover, FMM combined with parallel algorithms would be complicated and computationally expensive, which limits its extensive application in large-scale computing heavily (Yang and Stern 2017).
Hamilton–Jacobi equation, reaction action surface and the emergence of the force concept in chemical reaction dynamics
Published in Molecular Physics, 2018
The calculation of the MGAS, W(q0, q, n), by solving the GHJ equation (Equation 11) is done by adapting the FMM originally developed by Sethian [17–19] for solving eikonal equation where the wave front propagates by isotropic disturbances. Numerous applications of the FMM have been made over the past few years which include crystal growth, ray tracing, etching, robotic motion planning, computer vision, flame propagation and image processing. Inclusion of chemical reaction dynamics to this list, where the GAWF propagates by anisotropic disturbances defined by the potential energy, is relatively recent [15,16]. For details about the adaptation of the FMM to include chemical dynamics, we refer readers to the recent articles [15,16].
Modelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM
Published in Road Materials and Pavement Design, 2023
A. Dansou, S. Mouhoubi, C. Chazallon, M. Bonnet
The fast multipole method (FMM), introduced in Greengard and Rokhlin (1987), aims at improving the performance of boundary element analyses by circumventing the need to evaluate the kernel functions anew for each pair of boundary points encountered. This is achieved by introducing poles (Figure 3) at which contributions of clusters of points are gathered. The FMM rests on (i) decomposing the relative position vector as (Figure 3) and (ii) reformulating the kernel functions as truncated series of products of functions of the local position vectors . The cluster-wise treatment of contributions to integral operators is only valid for well-separated clusters. This motivates a recursive definition of such clusters using an octree-based partition of the space (smaller but nearer clusters becoming eligible to multipole expansions), which is the essence of the multi-level FMM used here. The multi-level elastostatic FMM enjoys as a result a computational complexity. The FMM implicitly splits the SGBEM matrix into , where gathers the contributions arising from multipole expansions and the close-range influence coefficients that have to be computed by traditional BEM quadrature (see Figure 4 for a schematic description). The matrix is of course not actually set up; rather, the FMM evaluates products that are used by an iterative solver (GMRES) applied to (10). Details of the FMM applied to elastostatic BIEs can be found in e.g. Yoshida (2001).