V. M. Jimenez-Fernandez

Biparameter Homotopy-based Direct Current Simulation of Multistable Circuits

The microelectronics area constantly demands better and improved circuit simulation tools. This is the reason that this article is to present a biparameter homotopy with automated stop criterion, which is applied to direct current simulation of multistable circuits. This homotopy possesses the following characteristics: symmetry axis, double bounding solution line, arbitrary initial and final points, and lessen the nonlinearities that exist in the circuit. Besides, this method will be exemplified and discussed by using a benchmark multistable circuit.

Exploring collision-free path planning by using homotopy continuation methods

Autonomous and semi-autonomous robots play significant roles in space and terrestrial exploration, even more in unfavorable and dangerous environments. Although recent advances allow robots to evolve in many such environments, one of the most important problems remains the establishment of collision-free trajectories in static or partially (temporal) static environments. This paper presents a different approach to address this problem, proposing a methodology based on homotopy continuation methods (HCM) capable of generating collision-free trajectories in two and three dimensions. The basic idea behind the proposal relies on the construction of a nonlinear equation representing the map of the environment, making it possible to apply HCM methods to obtain collision-free paths. A series of simulations are presented to show the effectiveness of the method avoiding circular, semi-rectangular, spherical shaped and semi-parallelepipeds obstacles.

Application-Specific Processor for Piecewise Linear Functions Computation

This paper presents an application specific processor architecture for the calculation of simplicial piecewise linear functions of up to six dimensions with 24-bit wide input words. The architecture, in particular registers and bus connections, is specifically designed for the task of simplicial piecewise linear computation. The parameters of the function are stored in an external 16 MB RAM memory. A proof-of-concept integrated circuit (that achieved first silicon success) was fabricated through MOSIS in a 4 mm × 4 mm 0.5 μm standard CMOS process using an automated design flow based on Synopsys and Cadence tools and the OSU standard cell library.

Nonlinearities distribution Laplace transform-homotopy perturbation method.

This article proposes non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM) to find approximate solutions for linear and nonlinear differential equations with finite boundary conditions. We will see that the method is particularly relevant in case of equations with nonhomogeneous non-polynomial terms. Comparing figures between approximate and exact solutions we show the effectiveness of the proposed method.

A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform-homotopy perturbation method

This article proposes Laplace Transform Homotopy Perturbation Method (LT-HPM) to find an approximate solution for the problem of an axisymmetric Newtonian fluid squeezed between two large parallel plates. After comparing figures between approximate and exact solutions, we will see that the proposed solutions besides of handy, are highly accurate and therefore LT-HPM is extremely efficient.

Applying an iterative-decomposed piecewise-linear model to find multiple operating points

This paper presents a new methodology for obtaining all the operating points in piecewise-linear (PWL) electrical networks. The networks can include independent and controlled voltage or current sources, resistors or conductances, and PWL elements. The nonlinear behavior of the PWL elements is graphically described by one-dimensional PWL curves. The mathematical representation of every PWL curve is given by an iterative and decomposed PWL model. The model is denominated as iterative because the representation of the segments belonging to the PWL curve depends on the value of one parameter included in the formulation. It is also denoted as decomposed because the two axis variables (x and y), in the PWL curve, are included separately in a system of linear equations. In order to optimize the process of searching all the operating points, the methodology is aided by a graphical procedure for identifying such constitutive element segments involved into the existence of any DC solution. By a numerical example here presented, it is possible to confirm the efficiency of the methodology.

Nonlinearities Distribution Homotopy Perturbation Method Applied to Solve Nonlinear Problems: Thomas-Fermi Equation as a Case Study

We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results.

Analytical Solutions for Systems of Singular Partial Differential-Algebraic Equations

This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations (SPDAEs). We will solve three examples to show that PSM method can be used to search for analytical solutions of SPDAEs. What is more, we will see that, in some cases, Pade posttreatment, besides enlarging the domain of convergence, may be employed in order to get the exact solution from the truncated series solutions of PSM.

Approximations for Large Deflection of a Cantilever Beam under a Terminal Follower Force and Nonlinear Pendulum

In theoretical mechanics field, solution methods for nonlinear differential equations are very important because many problems are modelled using such equations. In particular, large deflection of a cantilever beam under a terminal follower force and nonlinear pendulum problem can be described by the same nonlinear differential equation. Therefore, in this work, we propose some approximate solutions for both problems using nonlinearities distribution homotopy perturbation method, homotopy perturbation method, and combinations with Laplace-Pade posttreatment. We will show the high accuracy of the proposed cantilever solutions, which are in good agreement with other reported solutions. Finally, for the pendulum case, the proposed approximation was useful to predict, accurately, the period for an angle up to yielding a relative error of 0.01222747.

A Piecewise Linear Fitting Technique for Multivalued Two-dimensional Paths

Abstract This paper presents a curve-fitting technique for multivalued two-dimensional piecewise-linear paths. The proposed method is based on a decomposed formulation of the canonical piecewise linear model description of Chua and Kang. The path is treated as a parametric system of two position equations (x(k), y(k)), where k is an artificial parameter to map each variable (x and y) into an independent k-domain.