Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: In this article, a new analytical model of a gear pair with time varying mesh stiffness, viscous damping and sliding friction parameters is presented, where the excitation consists of three separate terms, namely the unloaded transmission error, time-invariant external torque and the periodically varying sliding friction force.
127 citations
TL;DR: In this paper, the authors define a finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface PDE posed on the boundary of the bulk domain.
Abstract: In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.
127 citations
TL;DR: It is shown that the use of minimum Frobenius norm quadratic models can improve the performance of direct-search methods and the minimization of the models within a trust region provides an enhanced search step.
Abstract: The goal of this paper is to show that the use of minimum Frobenius norm quadratic models can improve the performance of direct-search methods. The approach taken here is to maintain the structure of directional direct-search methods, organized around a search and a poll step, and to use the set of previously evaluated points generated during a direct-search run to build the models. The minimization of the models within a trust region provides an enhanced search step. Our numerical results show that such a procedure can lead to a significant improvement of direct search for smooth, piecewise smooth, and noisy problems.
126 citations
TL;DR: The relaxation derived using the multiparametric disaggregation technique (MDT) is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps.
Abstract: In this paper, we present the derivation of the multiparametric disaggregation technique (MDT) by Teles et al. (J. Glob. Optim., 2011) for solving nonconvex bilinear programs. Both upper and lower bounding formulations corresponding to mixed-integer linear programs are derived using disjunctive programming and exact linearizations, and incorporated into two global optimization algorithms that are used to solve bilinear programming problems. The relaxation derived using the MDT is shown to scale much more favorably than the relaxation that relies on piecewise McCormick envelopes, yielding smaller mixed-integer problems and faster solution times for similar optimality gaps. The proposed relaxation also compares well with general global optimization solvers on large problems.
126 citations
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TL;DR: In this article, the authors formulate the combinatorial Yamabe problem for piecewise flat metrics on manifolds and show that the flow either develops removable singularities or converges exponentially fast to a constant curvature metric.
Abstract: In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation.
We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.
126 citations