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Mathematical modeling and computer simulation of heat transfer in the liver play a key role in prediction of radio frequency ablation outcome. While multi-physics modeling should be used for accurate simulation of heat transfer in a highly perfused organ like the liver, the computational cost of such a problem could have a negative impact on the simulation time. This article extends a highly parallel finite element-based method to simulate the multi-physics heat transfer problem in the liver efficiently. A message passing interface (MPI)-based strong coupling method is developed and tested on cluster nodes. The scalability of this work is assessed by performing simulations on up to 1024 cores. [ABSTRACT FROM AUTHOR]

A novel implementation of a perfectly matched layer (PML) is presented for the truncation of finite-element time-domain (FETD) meshes containing electrically complex materials, exhibiting any combination of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Based on the complex coordinate stretching formulation of the PML, the presented technique yields an artificial absorbing layer whose matching condition is independent of material parameters. Moreover, by virtue of only modifying spatial derivatives, the incorporation of the PML into existing solvers for complex media is simple and straightforward. The resulting material-independent PML is incorporated into a nonlinear dispersive method for the vector wave equation, which leverages the $z$ transform and Newton-Raphson techniques to yield an implementation free from recursive convolutions, auxiliary differential equations, and linearizations. This permits the unprecedented truncation and attenuation of nonlinear phenomena, such as spatial and temporal solitons, within the FETD method. [ABSTRACT FROM AUTHOR]

In this paper, a mixed finite-element time-domain (FETD) method is presented for the simulation of electrically complex materials, including general combinations of linear dispersion, instantaneous nonlinearity, and dispersive nonlinearity. Using both edge and face elements, the presented method offers greater geometric flexibility than existing finite-difference time-domain (FDTD) implementations, and in contrast to existing nonlinear FETD methods, also incorporates both linear and nonlinear material dispersions. Dielectric nonlinearity is incorporated into the Crank–Nicolson mixed FETD formulation via a straightforward Newton–Raphson approach, for which the associated Jacobian is derived. Moreover, the dispersion is modeled via the Möbius $z$ transform method, yielding a simpler more general algorithm. The method’s accuracy and convergence are verified, and its capability demonstrated via the simulation of several nonlinear phenomena, including temporal and spatial solitons in two spatial dimensions. [ABSTRACT FROM AUTHOR]

A novel use of graphics processing units (GPUs) is presented in the execution of the dispersive finite-element time-domain (FETD) method, based upon the Möbius (bilinear) z -transform technique. By utilizing the immense computational power of modern GPUs via NVIDIA’s compute unified device architecture (CUDA) language, a narrowing of the performance gap, which currently exists between dispersive FETD methods and their non-dispersive counterparts, can be achieved, thus facilitating the study of a wider range of physical phenomena. An analysis of the z -transform dispersive FETD algorithm is presented in order to both identify dispersive overhead bottlenecks and determine its suitability to parallelization. Numerical studies are then undertaken to measure the performance increase as a function of simulation parameters, such as number of variables, the amount of dispersive material present, and floating point precision. [ABSTRACT FROM AUTHOR]

A new finite-difference time-domain (FDTD) formulation for modeling graphene is proposed, in which the graphene is modeled as a resistive sheet with a frequency-dependent conductivity. The formulation is first developed in the context of the vector wave finite-element time-domain and then reduced to the FDTD based on the equivalence between these two techniques. The obtained formulation is easy to implement and does not alter the original FDTD update equations. It can be applied to an existing FDTD code by simply adding a correction term to appropriate variables. One of the main contributions of this paper is analyzing the stability of the proposed formulation, which has not been done previously. [ABSTRACT FROM AUTHOR]

The finite-element Gaussian belief propagation (FGaBP) method, introduced recently, provides a powerful alternative to the conventional finite-element method solvers to efficiently utilize high-performance computing platforms. In this paper, we accelerate the FGaBP convergence by combining it with two methods based on residual minimization techniques, namely, the flexible generalized minimum residual and the iterant recombination method. The numerical results show considerable reductions in the total number of operations compared with the stand-alone FGaBP method, while maintaining the scalability features of FGaBP. [ABSTRACT FROM AUTHOR]

Two hybrid formulations combining the finite-element time-domain and the finite-difference time-domain methods are proposed for modeling dispersive materials. The methods are coupled to each other in a non-overlapping fashion using a new finite-element tearing and interconnecting algorithm. In addition, one of them is a fully unconditionally stable hybrid method, in which the time step is not restricted by a stability criterion. The effect of medium dispersion is considered using the Möbius transformation technique. The proposed methods are validated through numerical studies. [ABSTRACT FROM PUBLISHER]

The finite-element time-domain (FETD) solution of the vector wave equation (VWE) is directly extended to doubly dispersive media using the bilinear transform approach. The proposed formulation is quite general and flexible in the sense of material dispersion model in contrast to limitations of existing convolutional-based approaches. In addition, it can be implemented in a very straightforward and efficient manner. A stability analysis is performed that demonstrates the unconditional stability of the original formulation is preserved in the dispersive case. However, in order to avoid the well-known late-time instabilities that arise in this formulation, an alternative formulation is considered and extended to the dispersive case with the same approach. Several numerical examples are simulated to verify the validity and accuracy of the proposed formulations. The late-time stability of the alternative formulation is tested through a numerical example with 20 million time steps. The solution is completely free from late-time growth. [ABSTRACT FROM PUBLISHER]

A general framework to combine the finite-difference time-domain (FDTD) and the finite-element time-domain (FETD) formulations, both based on the vector wave equation, is proposed. In contrast to the existing stable hybrid FETD-FDTD, there is no transition layer between two subdomains. In addition, the stability of the proposed approach is analytically proved. This framework allows combining different FDTD and FETD formulations together. Particularly, a fully unconditionally stable hybrid method is proposed, which is proved to be energy conservative too. The key ingredient is a finite-element tearing and interconnecting method for electromagnetic problems with a new interface condition that preserves the stability of the numerical method in each region. Several numerical examples are considered in order to validate the proposed methods. The numerical results match with the reference solutions very well in all cases. [ABSTRACT FROM PUBLISHER]