Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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TL;DR: In this article, it was shown that if all the Jacobian determinants in the unbounded regions have the same sign, the equation f(x)= y has at least one solution and an algorithm was developed, which obtains one or more solutions in a finite number of steps.
Abstract: Nonlinear resistive networks can be characterized by the equation f(x)= y where f(x) is a continuous piecewise-linear mapping of Rn into itself. The n-dimensional Euclidean space is divided into a finite number of regions, and, in each region say region Rm, we can express f by J(m)x + w(m) where J(m) is a constant n × n Jacobian matrix and w(m) is a constant n-vector. In this paper we obtain the following results:
If all the Jacobian determinants in the unbounded regions have the same sign, the equation f(x)= y has at least one solution and an algorithm is developed, which obtains one or more solutions in a finite number of steps. The work represents a generalization of early work by Fujisawa, Kuh and Ohtsuki and relaxes the condition imposed on the function. For example, in the bounded regions, the Jacobian matrices can be singular and the sign of Jacobian determinants can be arbitrary.
45 citations
TL;DR: In this paper, the performance of two different approaches to solving the non-linear control allocation problem is presented, and the results show that solving the optimization problem using the simplex based approach is significantly faster than solving the same problem using a mixed-integer formulation.
Abstract: : The performance of two different approaches to solving the non-linear control allocation problem is presented. The non-linear control allocation problem is formulated using piecewise linear functions to approximate the control moments produced by a set of control effectors. when the control allocation problem is formulated as a piecewise linear program, an additional set of constraints enter into the problem formation. One approach is to introduce a set of binary variables to enforce these constraints. The result is a mixed- integer linear programming problem that can be solved using any branch-and-bound software. A second approach is to solve the piecewise linear programming problem using a modified simplex method. The simplex algorithm is modified to enforce a subset of the decision variables to enter into the basis only if certain conditions are met. We will show that solving the optimization problem using the simplex based approach is significantly faster than solving the same problem using a mixed-integer formulation. We will then compare the closed-loop performance of a re-entry vehicle using both approaches.
45 citations
TL;DR: In this article, the problem of identifying yield limits under piecewise linear yield surfaces and no local unstressing under increasing loads is shown to be amenable to the minimization of a convex quadratic function under linear and (nonconvex) complementarity constraints.
45 citations
TL;DR: On a Bakhvalov-type mesh the authors prove uniform convergence of a Galerkin finite element method with piecewise linear functions using a Clement quasi-interpolant as well as discretization error estimates in an energy norm.
45 citations
TL;DR: In this article, the classical Lindstedt-Poincare method is adapted to analyze the nonlinear normal modes of a piecewise linear system, where a simple two degrees of freedom, representing a beam with a breathing crack is considered.
Abstract: The classical Lindstedt-Poincare method is adapted to analyze the nonlinear normal modes of a piecewise linear system. A simple two degrees-of- freedom, representing a beam with a breathing crack is considered. The fundamental branches of the two modes and their stability are drawn by varying the severity of the crack, i.e., the level of nonlinearity. Re- sults furnished by the asymptotic method give insight into the mechanical behavior of the system and agree well with numerical results; the existence of super- abundant modes is proven. The unstable regions and the bifurcated branches are followed by a numerical procedure based on the Poincare map.
45 citations