Alternating Direction Implicit Finite-difference Method Research Articles

Efficient transient thermal analysis of chiplet heterogeneous integration

Thermal issue has significant effect on the reliability and performance of integrated circuits with the increase of integrated density, especially for 2.5-D and 3-D heterogeneous integration packaging systems. In this paper, a new highly efficient 3-D transient thermal distribution model has been constructed to capture the heat conduction behavior of multiple heat sources for chiplet heterogeneous integration (CHI) by improving the alternating direction implicit finite difference method (ADI-FDM). Firstly, a new floating optimization algorithm (FOA) of coefficient matrix assignment of heat conduction equation and a new boundary treatment utilizing virtual points are proposed to modify the ADI-FDM and improve the model efficiency of thermal analysis. Furthermore, a universal equivalent thermal conductivity model is also built up to describe the design features of through silicon via (TSV), bump and redistribution layer structures, which can enhance the flexibility of FDM-based thermal simulator. Compared with previous finite element analysis results, the present FOA-accelerated thermal model can not only obtain an accurate simulation result of temperature profile, but also give a simulation efficiency of 3.74 times faster than Douglas-Gunn approach for the coefficient matrix assignment. Finally, the effect of power consumption variation and the floorplaning effects of chiplets, TSVs and bumps on the temperature distribution are investigated in detail by utilizing the present FOA-based thermal solver. The simulated results of the present work demonstrate the viability and computational efficiency for temperature field optimization and indicate that the new proposed simulator is helpful in thermal analysis of packaging structures and can be adopted to assist in physical design optimization of 2.5-D CHI or 3-D heterogeneous stacked chips.

Efficient ADI schemes and preconditioning for a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients

In this paper, alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve a class of high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. We prove the unconditional stability and convergence rate of the ADI finite difference method provided that the diffusion coefficients satisfy the given conditions. For the linear system in each spatial direction, we establish a circulant approximate inverse preconditioner to accelerate the Krylov subspace method. In addition, we also use matrix-free algorithms and fast Fourier transforms (FFT) to speed up the solution of linear systems. Numerical experiments show the utility of the ADI method and the preconditioner.

An ADI finite difference method for the two-dimensional Volterra integro-differential equation with weakly singular kernel

ABSTRACT In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates in norm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis.

Hydromagnetic Natural Convection from a Horizontal Porous Annulus with Heat Generation or Absorption

This paper deals with a numerical study of free convection in a horizontal cylindrical annulus filled with a fluidsaturated porous medium in the presence of a transverse magnetic field and the heat generation or absorption effect. It is assumed that the inner and outer walls of the cylindrical annulus are maintained at constant temperatures Ti and To , respectively, as Ti > To . In addition to the heat equation, the model consists of the equation of motion under the Darcy law and Boussinesq approximation. The system of equations is solved numerically by the alternatingdirection implicit finite difference method. This investigation concerns the effects of the magnetic field inclination angle, Hartmann number, and the heat generation or absorption coefficient on heat transfer and the flow pattern. The results demonstrate that the heat transfer rate and flow regime depend mainly on the characteristics mentioned. The obtained data are presented graphically in terms of the streamlines and isotherms.

A 2nd-order ADI finite difference method for a 2D fractional Black–Scholes equation governing European two asset option pricing

Prices of options on two assets following two independent geometric Lévy processes are governed by a 2D fractional Black–Scholes (BS) equation. The discretization of the BS equation yields linear systems with dense system matrices and the numerical solution of them is computationally intensive. In this work, we develop a 2nd-order Crank–Nicolson Alternating Direction Implicit (ADI) method for solving these systems, based on a 2nd-order finite different technique proposed by us. A convergence theory for the method is established. Numerical results are presented to demonstrate the theoretical convergent rate of the ADI method and its computational efficiency.

Numerical Study of Transient and Steady-State Anisotropic Cylindrical Pin Fin Thermal Behavior

This paper deals with the transient thermal analysis of two-dimensional cylindrical anisotropic pin fin that contains tip convection and subjected to a prescribed temperature at the fin base. The heat conduction equation contains a dual second-order derivation, which precludes solving the equation by direct application of common exact methods. Therefore, an appropriate canonical mapping is selected as a solution to cancel the dual derivation of temperature in the mapped equations. The alternating-direction implicit finite difference method (ADI) performs the integration of the mapped equations in the novel space, which involve a complicate geometry. Applying the inverse spatial transformation provides transient temperature profile in the real geometry for full-field configuration. The established numerical code has been validated successfully with the analytical solutions of the usual fins (orthotropic and isotropic). The anisotropy effect is investigated by means of various contour plots of the temperature profile as well as heat transfer rate from the fin base and the effectiveness for different parameters of study (kr/kz, krz/kz , and Bir) in transient and steady-state heat conduction. The numerical code allows the study of the thermal behavior of anisotropic, orthotropic, and isotropic cylindrical pin fin according to the geometrical and physical parameters, as well as the thermal conditions to which the pin fin is subjected. A parametric study is performed in view to compare the thermal behavior of the various pin fin kinds submitted to the same conditions.

A characteristic ADI finite difference method for spatial-fractional convection-dominated diffusion equation

Based on characteristic method and shifted Grünwald fractional difference method, a characteristic finite difference method is proposed for solving the one/two/three-dimension spatial-fractional convection-dominated diffusion equation. The resulting schemes are first-order accuracy in time and second-order accuracy in space. For high-dimensional problems, alternating direction implicit (ADI) schemes are further proposed. The stability and convergence properties of these schemes are discussed. Numerical experiments are carried out to support the theoretical analysis, and some comparisons with the implicit upwind finite difference scheme are presented to show the effectiveness of the proposed method.

Numerical Solution of Unsteady Conduction Heat Transfer in Anisotropic Cylinders

This paper investigates a numerical solution of 2D transient heat conduction in an anisotropic cylinder, subjected to a prescribed temperature over the two end sections and a convective boundary condition over the whole lateral surface. The analysis of this anisotropic heat conduction problem is tedious because the corresponding partial differential equation contains a mixed-derivative. In order to overcome this difficulty, a linear coordinate transformation is used to reduce the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The alternating-direction implicit finite-difference method (ADI) is used to integrate the isotropic equation combined with boundary conditions. Inverse transformation provides profile temperature in the anisotropic cylinder for full-field configuration. The numerical code is validated by the analytical heat conduction solutions available in the literature such as transient isotropic solution and steady-state orthotropic solution. The aim of this paper is to study the effect of cross-conductivity on the temperature profile inside an axisymmetrical anisotropic cylinder versus time, radial Biot number (Bir), and principal conductivities. The results show that cross-conductivity promotes the effect of Bir according to the principal conductivities. Furthermore, the anisotropy increases the time required to achieve the steady-state heat conduction.

Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires computational work of O(N3) per time step and O(N2) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-, y-, z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a computational work count of O(NlogN) per iteration at each time step and a memory requirement of O(N). We also develop a fast multistep ADI finite difference method, which has a computational work count of O(Nlog2N) per time step and a memory requirement of O(NlogN). Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved computational cost and memory requirement. These numerical experiments show the utility of the fast method.

An Effective Finite Difference Method for Simulation of Bioheat Transfer in Irregular Tissues

A three-dimensional (3D) simulation of bioheat transfer is crucial to analyze the physiological processes and evaluate many therapeutic/diagnostic practices spanning from high to low temperature medicine. In this paper we develop an efficient numerical scheme for solving 3D transient bioheat transfer equations based on the alternating direction implicit finite-difference method (ADI-FDM). An algorithm is proposed to deal with the boundary condition for irregular domain which could capture accurately the complex boundary and reduce considerably the staircase effects. Furthermore, the local adaptive mesh technology is introduced to improve the computational accuracy for irregular boundary and the domains with large temperature gradient. The detailed modification to ADI-FDM is given to accommodate such special grid structure, in particular. Combination of adaptive-mesh technology and ADI-FDM could significantly improve the computational accuracy and decrease the computational cost. Extensive results of numerical experiments demonstrate that the algorithm developed in the current work is very effective to predict the temperature distribution during hyperthermia and cryosurgery. This work may play an important role in developing a computational planning tool for hyperthermia and cryosurgery in the near future.

An O( N log 2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second-order diffusion equations. Because of the nonlocal property of fractional differential operators, the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices. Consequently, the numerical solution of these methods often require computational work of O( N 3) per time step and memory of O( N 2) for where N is the number of grid points. In this paper we develop a fast alternating-direction implicit finite difference method for space-fractional diffusion equations in two space dimensions. The method only requires computational work of O( N log 2 N) per time step and memory of O( N), while retaining the same accuracy and approximation property as the regular finite difference method with Gaussian elimination. Our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations: To achieve the same accuracy, the new method has a significant reduction of the CPU time from more than 2 months and 1 week consumed by a traditional finite difference method to 1.5 h, using less than one thousandth of memory the standard method does. This demonstrates the utility of the method.

Modeling credit spreads with the Cheyette model and its application to credit default swaptions

In this paper we apply Cheyette's Markov representation of the Heath-Jarrow-Morton framework to the modeling of stochastic credit spreads. As an application of this framework, the volatility of the credit spread process is modeled by considering the constant elasticity of variance approach of Ritchken and Sankarasubramanian and the Andersen-Andreasen displaced approach. To examine the practicability of this approach, we calibrate the model to market prices of credit default swaptions. Thereby we use Monte Carlo simulation and the alternating direction implicit finite-difference method.

Stability analysis of the Mur's absorbing boundary condition in the alternating direction implicit finite-difference method

The stability analysis of the Mur's first order absorbing boundary condition (ABC) in the alternating direction implicit finite-difference time-domain (ADI-FDTD) method is presented. The theoretical stability analysis of this scheme is studied by deriving the amplification matrix. The effect of wave propagation direction on the stability of this scheme is investigated. From the stability analysis, it is found that the Mur's first order ABC in the ADI-FDTD method will be unstable. The instability of the scheme is validated by means of actual numerical simulations.

Scanning electrochemical microscopy. Theory of the feedback mode for hemispherical ultramicroelectrodes: steady-state and transient behavior

This contribution represents the first comprehensive attempt to treat complex geometry configurations of the scanning electrochemical microscope (SECM) using the alternating direction implicit finite difference method (ADIFDM). Specifically, ADIFDM is used to simulate the steady-state as well as the transient (chronoamperometric) behavior of a hemispherical ultramicroelectrode (UME) tip of the SECM. The feedback effect in this configuration is less pronounced as compared with a disk-shaped UME system. The differences between the two systems are discussed. Analytical approximations for the steady-state behavior and for characteristic features of the transient behavior are suggested. Finally, experimental feedback currents measured above a conductor and an insulator are in excellent agreement with the theory.

Comprehensive treatment of sphere-cap microelectrodes (SCMs) using digital simulations

The alternating direction implicit finite difference method (ADIFDM) is used to simulate the steady-state as well as the transient (chronoamperometric) behavior of sphere-cap microelectrodes (SCMs). The necessary steps to implement the ADIFDM for solving the rather complex geometry involved are described. The results for the steady-state behavior are in excellent agreement with a previously suggested analytical solution making the ADIFDM an excellent tool to develop a comprehensive theory for SCMs.

Simulation of Gas Transport in an Intravenacaval Blood-Gas Exchange device (IVOX)©

A mathematical model of a novel intravascular hollow-fiber gas exchange device, called IVOX©, is developed using a Krogh cylinderlike approach wherein a unit structure is assumed to comprise of a single fiber with gas flowing through its lumen and surrounded by another coaxial cylinder of blood flowing in the opposite direction. Species mass balances on O2 and CO2 result in a nonlinear coupled set of convective-difTusion parabolic partial differential equations that are solved numerically using an alternating direction implicit finite-difference method. Computed results indicate the presence of a large resistance to gas transport on the external (blood) side of the hollow fiber exchanger. Increasing gas flow through the device was found to favor CO2 removal but not O2 addition to blood. On the other hand, increasing blood flow over the device favored both CO2 removal as well as O2 addition. Neglecting the vacuum pressure gradient on the gas side results in an underestimate of CO2 transport but an overestimate of O2 transport. The exchange rates of CO2 and O2 were found to increase directly with the transmural Pco2 gradient imposed across the device. This strongly suggests that CO2 removal may be enhanced in a clinical setting under conditions of permissive hypercapnia (controlling respiratory acidosis by bicarbonate infusion). Fiber crimping was studied indirectly; improved blood mixing produced by the crimped fibers was mimicked by increasing species diffusivity in blood. The resulting lowered blood-side resistance favored species transfer.

Three-dimensional transient heat transfer from a buried pipe—I. Laminar flow

A model is presented for heat transfer from a fluid in laminar flow inside a buried horizontal pipe. The top surface of the soil in which the pipe is buried is horizontal and flat and is kept at a temperature lower than the soil and the liquid temperatures. The problem is modelled as a transient three-dimensional process in cylindrical geometry. The solution is obtained using a string-intersected-boundary formulation and a three-level alternating-direction implicit finite difference method. The resulting equations were solved on a CDC Cyber 205 supercomputer. The validity of the numerical scheme has been verified by comparison with an analytical solution for a simplified problem.

Propagating beam analysis by alternating-direction implicit finite-difference method

The three-dimensional beam propagation problem expressed by the Fresnel equation is analyzed by the beam-propagation method using an alternating-direction implicit finite-difference method (ADI-BPM). First, following the formulation of the finite-difference method, the equations to be analyzed are rearranged. The fundamental and higher-order mode propagation problems in a step index fiber are treated, and the accuracy and advantages of the ADI-BPM are investigated. It is found that the computing time is shorter and the step length in the propagation direction can be made larger in the ADI-BPM than in the beam-propagation method based on the fast Fourier transform (FFT-BPM). As another advantage in the ADI-BPM, the usefulness of the transparent boundary condition is described. It is shown that the spreading problem of a Gaussian beam can be analyzed without setting up the absorbing functions needed in the FFT-BPM. Further, as an example to prove the effectiveness of the ADI-BPM for the analysis of the problem in which the guided mode and the radiation mode are mixed, the transmission efficiency is evaluated in the case where an axial offset exists in the fiber junctions. The effect of the coherent coupling is studied.

Models of the rhizosphere

A mathematical model is outlined which is capable of simulating the radial and vertical distribution of soluble carbon, and of microbial biomass and necromass around a root growing through the soil. An alternating-direction implicit finite difference method is used to simulate the movement of soluble C by diffusion and convection in the rhizosphere cylinder surrounding the root. Results are presented which suggest that microbial populations in the rhizosphere of a growing root may vary considerably with distance along the root with the precise distribution of soluble C and biomass depending on the pattern of exudate release along the root length. © 1991 Kluwer Academic Publishers.

Models of the rhizosphere

A mathematical model is outlined which is capable of simulating the radial and vertical distribution of soluble carbon, and of microbial biomass and necromass around a root growing through the soil. An alternating-direction implicit finite difference method is used to simulate the movement of soluble C by diffusion and convection in the rhizosphere cylinder surrounding the root. Results are presented which suggest that microbial populations in the rhizosphere of a growing root may vary considerably with distance along the root with the precise distribution of soluble C and biomass depending on the pattern of exudate release along the root length.