IMPROVED ALTERNATING-DIRECTION IMPLICIT METHOD FOR SOLVING TRANSIENT THREE-DIMENSIONAL HEAT DIFFUSION PROBLEMS

We propose a multigrid Alternating-Direction Implicit (ADI) method for solving Maxwell's equations in this paper. This method is based on the interpretation of the ADI method as an iterative solver of the Crank-Nicolson (CN) scheme. By introducing a special solving procedure for the residual equation within the ADI framework, multigrid schemes are incorporated into the iterative ADI method. The accuracy and efficiency of the proposed multigrid ADI method are further demonstrated by numerical examples.

This paper describes a circuit-oriented leapfrog alternating direction implicit (ADI) method applied to a simulation of a power delivery network (PDN). An equivalent circuit model of the PDN is derived by using a recently-proposed conformal modeling technique, in which orthogonal meshes are used along with a treatment to improve the accuracy of the model. Since the conformal equivalent circuit model has a grid-like topology, it is suitable for direction-based algorithms such as an ADI method and the leapfrog ADI method. An advantage of the ADI-based method is that they can use a relatively-large time step size and reduce the total number of time steps in a transient simulation. In addition, the leapfrog ADI method is more efficient than the existing ADI method because its time marching procedure requires less updating processes. As a result, the conformal equivalent circuit model combined with the leapfrog ADI method can be accurate and efficient in modeling and analysis of the PDN.

The approximate-factorisation-splitting (CNAFS) method as an efficient implementation of the Crank-Nicolson scheme for solving the three-dimensional Maxwell equations in the time domain, using much less CPU time and memory than a direct implementation, is presented. At each time step, the CNAFS method solves tridiagonal matrices successively instead of solving a huge sparse matrix. It is shown that CNAFS is unconditionally stable and has much smaller anisotropy than the alternating-direction implicit (ADI) method, though the numerical dispersion is the same as in the ADI method along the axes. In addition, for a given mesh density, there will be one value of the Courant number at which the CNAFS method has zero anisotropy, whereas the Crank-Nicolson scheme always has anisotropy. Analysis shows that both ADI and CNAFS have time step-size limits to avoid numerical attenuation, although both are still unconditionally stable beyond their limit.

ABSTRACTHeat transfer at microscale plays an important role in microtechnology applications. First, this paper is concerned with the numerical solution of a two-dimensional (2D) heat transfer equation at microscale through a second-order alternating direction implicit (ADI) method. By the discrete energy method, it is shown that the ADI solution converges to the exact solution with a convergence order of in L2-norm. Second, the ADI method is generalized to numerically solve the three-dimensional (3D) problem, and provides numerical solutions of order two in both time and space. Finally, numerical results testify the temporal and spatial accuracy of the ADI method.

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.

A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems

A matched Peaceman–Rachford ADI method for solving parabolic interface problems

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

SummaryThis paper presents fast alternating direction implicit (FADI) method for efficient transient thermal simulation of integrated circuits. The FADI method is formulated from Peaceman–Rachford's ADI and Douglas–Gunn's ADI methods. The update procedure of the proposed method has basic implicit form that features derivative‐free right‐hand side and hence, better efficiency and conciseness. Subsequently, through the basic implicit form of FADI method, the relationship between classical Peaceman–Rachford's and Douglas–Gunn's ADI methods can be clarified and elucidated in detail. A unified boundary condition that can cater to common kinds of boundary conditions in thermal simulation is also introduced. To further accelerate FADI method, the graphics processing unit is also utilized through Compute Unified Device Architecture implementation. It is shown that high efficiency gain can be achieved using the proposed FADI method through large time step size and data parallelism, while maintaining stability and good accuracy. As numerical illustration, an integrated circuit structure with microchannel cooling is demonstrated. Numerical results further ascertain the cooling effect of the microchannels. Copyright © 2015 John Wiley & Sons, Ltd.

A simultaneous estimation of three boundary conditions of heat conduction problem is proposed by numerical approach. A finite-difference method is used to discretize the governing equations. The aim is to estimate the evolution of the distributions of the unknown heat fluxes from the transient temperature histories taken with several sensors sensors inside a two-dimensional sheet. The temperatures are known at three lines in the finite body. The estimation algorithm of this inverse heat conduction problem is based on the iterative regularization method and on the conjugate gradient method. The alternative direction implicit method is used to solve the direct, the adjoin and the variation problems. For each boundary condition, a descent parameter is computed. An optimal choice of the vector of the descent parameters is used in this study and shows an increase in the convergence rate. All numerical simulations are performed for two-dimensional linear heat conduction problem. The temperatures are given with measurement errors and the iterative process is stopped in accordance with the residua criterion. Numerical results are presented in this study. The temperature and heat fluxes graphs are smoothed. The accuracy and efficiency of the inverse analysis for simultaneously estimating the heat fluxes and the temperature is examined by several cases. Finally, the effects of sensor position and the magnitude of measurement error on the inverse solutions are discussed. Numerical results for some representative cases prove that heat fluxes and temperature can be predicted well by this method. In this paper, simulation results are presented and numerical performance of the method will be discussed.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

We present a numerical study of transient natural convection in a rectangular open thermosyphon having asymmetric thermal boundary conditions. One vertical wall of the thermosyphon is either heated by constant heat flux (“warmup”) or cooled by convection to the surroundings (“cooldown”). The top of the thermosyphon is open to a large reservoir of fluid at constant temperature. The vorticity, energy, and stream-function equations are solved by finite differences on graded mesh. The ADI method and iteration with overrelaxation are used. We find that the thermosyphon performs quite differently during cooldown compared with warmup. In cooldown, flows are mainly confined to the thermosyphon with little momentum and heat exchange with the reservoir. For warmup, the circulation resembles that for a symmetrically heated thermosyphon where there is a large exchange with the reservoir. The difference is explained by the temperature distributions. For cooldown, the fluid becomes stratified and the resulting stability reduces motion. In contrast, the transient temperature for warmup does not become stratified but generally exhibits the behavior of a uniformly heated vertical plate. For cooldown and Ra > 104, time-dependent heat transfer is predicted by a closed-form expression for one-dimensional conduction, which shows that Nu → Bi1/2/A in the steady-state limit. For warmup, transient heat transfer behaves as one-dimensional conduction for early times and at steady state and for Ra* ≥ 105, can be approximated as that for a uniformly heated vertical plate.

In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL$^T$ formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient $O(N)$ convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers which are unconditionally stable, accurate of order $2P$, and require $O(P^d N)$ operations per time step, where $N$ is the number of spatial points and $d$ the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs and can also be extended to parabolic PDEs.

We consider the initial boundary value problem for a fractional evolution integro-differential equation with a weakly singular kernel arising from viscoelastic rods and membranes in a bounded smooth domain. In our numerical scheme, an arbitrary-order orthogonal spline collocation (OSC) method is used for spatial approximation and combined with the Crank-Nicolson (CN) alternating direction implicit (ADI) method for time approximation, and the integral term is approximated via the product integration (PI) rule. The detailed stability and convergence of the CN ADI OSC method in norm are proved. Finally, some numerical simulations are provided to illustrate our theoretical error estimates. In addition, the comparison between the non-uniform mesh and the uniform mesh shows the effectiveness of the non-uniform mesh when the solution is not sufficiently smooth.

Large degree-of-freedom real-time adaptive optics (AO) control requires reconstruction algorithms that are computationally efficient and readily parallelized for hardware implementation. In particular, we find the wave-front reconstruction for the Hudgin and Fried geometry can be cast into a form of the well-known Sylvester equation using the Kronecker product properties of matrices. We derive the filters and inverse filtering formulas for wave-front reconstruction in two-dimensional (2-D) Discrete Cosine Transform (DCT) domain for these two geometries using the Hadamard product concept of matrices and the principle of separable variables. We introduce a recursive filtering (RF) method for the wave-front reconstruction on an annular aperture, in which, an imbedding step is used to convert an annular-aperture wave-front reconstruction into a squareaperture wave-front reconstruction, and then solving the Hudgin geometry problem on the square aperture. We apply the Alternating Direction Implicit (ADI) method to this imbedding step of the RF algorithm, to efficiently solve the annular-aperture wave-front reconstruction problem at cost of order of the number of degrees of freedom, O(n). Moreover, the ADI method is better suited for parallel implementation and we describe a practical real-time implementation for AO systems of order 3,000 actuators.