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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 24 Feb 2024 06:03:38 GMT2024-02-24T06:03:38ZProper generalized decomposition solutions within a domain decomposition strategy
http://hdl.handle.net/10985/13823
Proper generalized decomposition solutions within a domain decomposition strategy
HUERTA, Antonio; NADAL, Enrique; CHINESTA, Francisco
Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane–out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/138232018-01-01T00:00:00ZHUERTA, AntonioNADAL, EnriqueCHINESTA, FranciscoDomain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane–out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.On the physical interpretation of fractional diffusion
http://hdl.handle.net/10985/13278
On the physical interpretation of fractional diffusion
NADAL, Enrique; ABISSET-CHAVANNE, Emmanuelle; CUETO, Elias; CHINESTA, Francisco
Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10985/132782018-01-01T00:00:00ZNADAL, EnriqueABISSET-CHAVANNE, EmmanuelleCUETO, EliasCHINESTA, FranciscoEven if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.Real-Time Path Planning Based on Harmonic Functions under a Proper Generalized Decomposition-Based Framework
http://hdl.handle.net/10985/20395
Real-Time Path Planning Based on Harmonic Functions under a Proper Generalized Decomposition-Based Framework
MONTÉS, Nicolas; CHINESTA, Francisco; MORA, Marta C.; FALCÓ, Antonio; HILARIO, Lucia; ROSILLO, Nuria; NADAL, Enrique
This paper presents a real-time global path planning method for mobile robots using harmonic functions, such as the Poisson equation, based on the Proper Generalized Decomposition (PGD) of these functions. The main property of the proposed technique is that the computational cost is negligible in real-time, even if the robot is disturbed or the goal is changed. The main idea of the method is the off-line generation, for a given environment, of the whole set of paths from any start and goal configurations of a mobile robot, namely the computational vademecum, derived from a harmonic potential field in order to use it on-line for decision-making purposes. Up until now, the resolution of the Laplace or Poisson equations has been based on traditional numerical techniques unfeasible for real-time calculation. This drawback has prevented the extensive use of harmonic functions in autonomous navigation, despite their powerful properties. The numerical technique that reverses this situation is the Proper Generalized Decomposition. To demonstrate and validate the properties of the PGD-vademecum in a potential-guided path planning framework, both real and simulated implementations have been developed. Simulated scenarios, such as an L-Shaped corridor and a benchmark bug trap, are used, and a real navigation of a LEGO®MINDSTORMS robot running in static environments with variable start and goal configurations is shown. This device has been selected due to its computational and memory-restricted capabilities, and it is a good example of how its properties could help the development of social robots.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10985/203952021-01-01T00:00:00ZMONTÉS, NicolasCHINESTA, FranciscoMORA, Marta C.FALCÓ, AntonioHILARIO, LuciaROSILLO, NuriaNADAL, EnriqueThis paper presents a real-time global path planning method for mobile robots using harmonic functions, such as the Poisson equation, based on the Proper Generalized Decomposition (PGD) of these functions. The main property of the proposed technique is that the computational cost is negligible in real-time, even if the robot is disturbed or the goal is changed. The main idea of the method is the off-line generation, for a given environment, of the whole set of paths from any start and goal configurations of a mobile robot, namely the computational vademecum, derived from a harmonic potential field in order to use it on-line for decision-making purposes. Up until now, the resolution of the Laplace or Poisson equations has been based on traditional numerical techniques unfeasible for real-time calculation. This drawback has prevented the extensive use of harmonic functions in autonomous navigation, despite their powerful properties. The numerical technique that reverses this situation is the Proper Generalized Decomposition. To demonstrate and validate the properties of the PGD-vademecum in a potential-guided path planning framework, both real and simulated implementations have been developed. Simulated scenarios, such as an L-Shaped corridor and a benchmark bug trap, are used, and a real navigation of a LEGO®MINDSTORMS robot running in static environments with variable start and goal configurations is shown. This device has been selected due to its computational and memory-restricted capabilities, and it is a good example of how its properties could help the development of social robots.