A. Falcó

Numerical strategies for the Galerkin–proper generalized decomposition method

Abstract The Proper Generalized Decomposition or, for short, PGD is a tensor decomposition based technique to solve PDE problems. It reduces calculation and storage cost drastically and presents some similarities with the Proper Orthogonal Decomposition, for short POD. In this work, we propose an efficient implementation to improve the convergence of the PGD, toward the numerical solution of a discretized PDE problem, when the associated matrix is Laplacian-like.

Improving Computational Efficiency in LCM by Using Computational Geometry and Model Reduction Techniques

LCM simulation is computationally expensive because it needs an accurate solution of flowequations during the mold filling process. When simulating large computing times are not compatiblewith standard optimization techniques (for example for locating optimally the injection nozzles)or with process control that in general requires fast decision-makings. In this work, inspired by theconcept of medial axis, we propose a numerical technique that computes numerically approximatedistance fields by invoking computational geometry concepts that can be used for the optimal locationof injection nozzles in infusion processes. On the other hand we also analyze the possibilities thatmodel order reduction offers to fast and accurate solutions of flow models in mold filling processes.

Real-time Bézier Trajectory Deformation for Potential Fields planning methods

This article presents a new technique for obtaining a flexible trajectory based on the deformation of a Bézier curve through a field of vectors. This new technique is called Bézier Trajectory Deformation (BTD). The trajectory deformation is computed with a constrained optimization method (Lagrange Multipliers Theorem). A linear system is solved to achieve the result. As a consequence, the deformed trajectory is computed in a few milliseconds. In addition, the linear system can be solved offline if the Bézier curve order is maintained constant during the movement of the robot, which is the common case. This technique can be combined with any collision avoidance algorithm that produces a field of vectors. In particular, it has been developed for artificial potential field methods. BTD is combined with a recently proposed PF method, the Potential Field Projection method (PFP). The resulting technique is tested in dynamic environment showing its real-time performance and its efficacy for obstacle avoidance.

Towards a 2.5D geometric model in mold filling simulation

Resin Infusion (RI) process is frequently used for large composite parts production. This Liquid Composite Molding method uses vacuum pressure to shape a plastic bag as a counter mold. Once a complete vacuum is achieved, the resin is sucked into a dry preform textile laminate via placed tubing. In this note we introduce a 2.5D model for a Liquid Composite Molding LCM process starting from a recent one introduced by Besson and Poussin in Besson and Pousin (2005) for the Resin Transfer Molding RTM process. Moreover for 2.5D models defined over quadrilateral reference domains, we show the effectiveness of the use of the Proper Generalized Decomposition. Finally, we propose a procedure, for a particular class of 2.5D model, in order to perform a numerical simulation of a mold filling process.

A tensor optimization algorithm for Bézier Shape Deformation

In this paper we propose a tensor based description of the Bezier Shape Deformation (BSD) algorithm, denoted as T-BSD. The BSD algorithm is a well-known technique, based on the deformation of a Bezier curve through a field of vectors. A critical point in the use of real-time applications is the cost in computational time. Recently, the use of tensors in numerical methods has been increasing because they drastically reduce computational costs. Our formulation based in tensors T-BSD provides an efficient reformulation of the BSD algorithm. More precisely, the evolution of the execution time with respect to the number of curves of the BSD algorithm is an exponentially increasing curve. As the numerical experiments show, the T-BSD algorithm transforms this evolution into a linear one. This fact allows to compute the deformation of a Bezier with a much lower computational cost.

On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems

Abstract In this study, we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition (PGD). The PGD is a novel technique that was developed recently for solving problems with high dimensions, and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method. This new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the PGD can help in the change to this paradigm. The main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum. We prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions. In particular, we illustrate this methodology based on a robot path planning problem. This is one of the common tasks when designing the trajectory or path of a mobile robot. The construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map.

A general framework for a class of non-linear approximations with applications to image restoration

Abstract In this paper, we establish sufficient conditions for the existence of optimal non-linear approximations to a linear subspace generated by a given weakly-closed (non-convex) cone of a Hilbert space. Most non-linear problems have difficulties to implement good projection-based algorithms due to the fact that the subsets, where we would like to project the functions, do not have the necessary geometric properties to use the classical existence results (such as convexity, for instance). The theoretical results given here overcome some of these difficulties. To see this we apply them to a fractional model for image deconvolution. In particular, we reformulate and prove the convergence of a computational algorithm proposed in a previous paper by some of the authors. Finally, some examples are given.

A Computational Approach Based on Flow Front Shape Dynamic Behavior for the Process Characterization during Filling in Liquid Resin Infusion

Resin Infusion (RI) process is one of the common techniques used in the industry for large composite parts production. This technique uses vacuum pressure to drive the resin into a laminate. Preform is laid dry into the mold and the vacuum is applied before the resin is introduced. Once a complete vacuum is achieved, resin is sucked into the laminate via placed tubing. Fig.1 shows a diagram of this process.An appropriate modeling of flow front’s shapes constrained by LRI process during filling can be based on the continuous deformation of the vent oriented flow pattern due to the driving pressure from the inlet. One of the main objectives is that the flow achieves the contour vent uniformly to avoid pressure drop and ensuring complete filling. In LRI, the flow front shape progression is mainly conditioned by the initial arrangement of the injection line allocation and the permeability of the preform that can evolve along the mold