A multi-microprocessor architecture for solving partial differential equations

This paper presents the architecture of a fault-tolerant, special-purpose multi-microprocessor system for solving Partial Differential Equations (PDEs). The modular nature of the architecture allows the use of hundreds of Processing Elements (PEs) for high throughput. Its performance is evaluated by both analytical and simulation methods. The results indicate that the system can achieve high operation rates and is not sensitive to inter-processor communication delay.

A Method of Characteristics for Solving Two-dimensional Unsteady Flow Problems

The aim of this investigation is to evolve a method of solving two-dimensional unsteady flow problems by the method of characteristics. This involves the reduction of the given system of equations to an equivalent system where only interior derivatives occur on a characteristic surface. From this system, four special bicharacteristic directional derivatives are chosen. A finite difference scheme is prescribed for solving the equations. General rectangular lattices are also considered. As an example, we investigate the propagation of an initial pressure distribution in a medium at rest.

Analytical method for solving a set of inhomogeneous infinite simultaneous equations

It is shown that a method based on the principle of analytic continuation can be used to solve a set of inhomogeneous infinite simultaneous equations encountered in the analysis of surface acoustic wave propagation along the periodically perturbed surface of a piezoelectric medium.

Analytical method for solving a set of infinite simultaneous equations

It is shown that a method based on the principle of analytic continuation can be used to solve a set of infinite simultaneous equations encountered in solving for the electric field of a periodic electrode structure.

The alpha method for solving differential algebraic inequality (DAI) systems

This paper describes an algorithm for ``direct numerical integration'' of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation's (DAE) Jacobian and this difficulty is addressed by a regularization property of the alpha method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

Solving MDPs using two-timescale simulated annealing with multiplicative weights

We develop extensions of the Simulated Annealing with Multiplicative Weights (SAMW) algorithm that proposed a method of solution of Finite-Horizon Markov Decision Processes (FH-MDPs). The extensions developed are in three directions: a) Use of the dynamic programming principle in the policy update step of SAMW b) A two-timescale actor-critic algorithm that uses simulated transitions alone, and c) Extending the algorithm to the infinite-horizon discounted-reward scenario. In particular, a) reduces the storage required from exponential to linear in the number of actions per stage-state pair. On the faster timescale, a 'critic' recursion performs policy evaluation while on the slower timescale an 'actor' recursion performs policy improvement using SAMW. We give a proof outlining convergence w.p. 1 and show experimental results on two settings: semiconductor fabrication and flow control in communication networks.

A parallel multistep predictor-corrector algorithm for solving ordinary differential equations

In this paper we give a generalized predictor-corrector algorithm for solving ordinary differential equations with specified initial values. The method uses multiple correction steps which can be carried out in parallel with a prediction step. The proposed method gives a larger stability interval compared to the existing parallel predictor-corrector methods. A method has been suggested to implement the algorithm in multiple processor systems with efficient utilization of all the processors.

A study of symbolic processing and computational aspects in helicopter dynamics

Even research models of helicopter dynamics often lead to a large number of equations of motion with periodic coefficients; and Floquet theory is a widely used mathematical tool for dynamic analysis. Presently, three approaches are used in generating the equations of motion. These are (1) general-purpose symbolic processors such as REDUCE and MACSYMA, (2) a special-purpose symbolic processor, DEHIM (Dynamic Equations for Helicopter Interpretive Models), and (3) completely numerical approaches. In this paper, comparative aspects of the first two purely algebraic approaches are studied by applying REDUCE and DEHIM to the same set of problems. These problems range from a linear model with one degree of freedom to a mildly non-linear multi-bladed rotor model with several degrees of freedom. Further, computational issues in applying Floquet theory are also studied, which refer to (1) the equilibrium solution for periodic forced response together with the transition matrix for perturbations about that response and (2) a small number of eigenvalues and eigenvectors of the unsymmetric transition matrix. The study showed the following: (1) compared to REDUCE, DEHIM is far more portable and economical, but it is also less user-friendly, particularly during learning phases; (2) the problems of finding the periodic response and eigenvalues are well conditioned.

Clan-based evolutionary approach for solving control problems

This paper presents a dan-based evolutionary approach for solving control problems. Three selected control problems, viz. linear-quadratic, harvest, and push-cart problems, are solved using the proposed approach. Results are compared with those of the evolutionary programming (EP) approach. In most of the cases, the proposed approach is successful in obtaining (near) optimal solutions for these selected problems.

Solving linear programming problems exactly

A linear programming problem in an inequality form having a bounded solution is solved error-free using an algorithm that sorts the inequalities, removes the redundant ones, and uses the p-adic arithmetic. (C) Elsevier Science Inc., 1997

Machine-tool selection and operation allocation in FMS: solving a fuzzy goal-programming model using a genetic algorithm

The decision-making process for machine-tool selection and operation allocation in a flexible manufacturing system (FMS) usually involves multiple conflicting objectives. Thus, a fuzzy goal-programming model can be effectively applied to this decision problem. The paper addresses application of a fuzzy goal-programming concept to model the problem of machine-tool selection and operation allocation with explicit considerations given to objectives of minimizing the total cost of machining operation, material handling and set-up. The constraints pertaining to the capacity of machines, tool magazine and tool life are included in the model. A genetic algorithm (GA)-based approach is adopted to optimize this fuzzy goal-programming model. An illustrative example is provided and some results of computational experiments are reported.

Solving a part classification problem using simulated annealing-like hybrid

Part classification and coding is still considered as laborious and time-consuming exercise. Keeping in view, the crucial role, which it plays, in developing automated CAPP systems, the attempts have been made in this article to automate a few elements of this exercise using a shape analysis model. In this study, a 24-vector directional template is contemplated to represent the feature elements of the parts (candidate and prototype). Various transformation processes such as deformation, straightening, bypassing, insertion and deletion are embedded in the proposed simulated annealing (SA)-like hybrid algorithm to match the candidate part with their prototype. For a candidate part, searching its matching prototype from the information data is computationally expensive and requires large search space. However, the proposed SA-like hybrid algorithm for solving the part classification problem considerably minimizes the search space and ensures early convergence of the solution. The application of the proposed approach is illustrated by an example part. The proposed approach is applied for the classification of 100 candidate parts and their prototypes to demonstrate the effectiveness of the algorithm. (C) 2003 Elsevier Science Ltd. All rights reserved.

Smooth DMS-FEM: A new approach to solving nearly incompressible nonlinear elasto-static problems

The DMS-FEM, which enables functional approximations with C(1) or still higher inter-element continuity within an FEM-based meshing of the domain, has recently been proposed by Sunilkumar and Roy [39,40]. Through numerical explorations on linear elasto-static problems, the method was found to have conspicuously superior convergence characteristics as well as higher numerical stability against locking. These observations motivate the present study, which aims at extending and exploring the DMS-FEM to (geometrically) nonlinear elasto-static problems of interest in solid mechanics and assessing its numerical performance vis-a-vis the FEM. In particular, the DMS-FEM is shown to vastly outperform the FEM (presently implemented through the commercial software ANSYS (R)) as the former requires fewer linearization and load steps to achieve convergence. In addition, in the context of nearly incompressible nonlinear systems prone to volumetric locking and with no special numerical artefacts (e.g. stabilized or mixed weak forms) employed to arrest locking, the DMS-FEM is shown to approach the incompressibility limit much more closely and with significantly fewer iterations than the FEM. The numerical findings are suggestive of the important role that higher order (uniform) continuity of the approximated field variables play in overcoming volumetric locking and the great promise that the method holds for a range of other numerically ill-conditioned problems of interest in computational structural mechanics. (C) 2011 Elsevier Ltd. All rights reserved.

Dilations of Gamma-contractions by solving operator equations

For a contraction P and a bounded commutant S of P. we seek a solution X of the operator equation S - S*P = (1 - P* P)(1/2) X (1 - P* P)(1/2) where X is a bounded operator on (Ran) over bar (1 - P* P)(1/2) with numerical radius of X being not greater than 1. A pair of bounded operators (S, P) which has the domain Gamma = {(z(1) + z(2), z(2)): vertical bar z(1)vertical bar < 1, vertical bar z(2)vertical bar <= 1} subset of C-2 as a spectral set, is called a P-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Gamma-contraction (S, P). This allows us to construct an explicit Gamma-isometric dilation of a Gamma-contraction (S, P). We prove the other way too, i.e., for a commuting pair (S, P) with parallel to P parallel to <= 1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S, P) is a Gamma-contraction. We show that for a pure F-contraction (S, P), there is a bounded operator C with numerical radius not greater than 1, such that S = C + C* P. Any Gamma-isometry can be written in this form where P now is an isometry commuting with C and C. Any Gamma-unitary is of this form as well with P and C being commuting unitaries. Examples of Gamma-contractions on reproducing kernel Hilbert spaces and their Gamma-isometric dilations are discussed. (C) 2012 Elsevier Inc. All rights reserved.