Computer methods in applied mechanics and engineering

Vol. 295, p. 127-149

DOI: 10.1016/j.cma.2015.03.026

Date of publication: 2015-10-01

Abstract:

Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies. The Proper Generalized Decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a Perfectly Matched Layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an efficient higher-order projection scheme. Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov-Galerkin approach to construct the separated basis. Moreover, the efficiency of the higherorder projection scheme is demonstrated and compared with the higher-order singular value decomposition.]]>

International journal for numerical methods in engineering

Vol. 103, num. 10, p. 737-758

DOI: 10.1002/nme.4909

Date of publication: 2015-09-07

Abstract:

The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The proper generalized decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., Parametric solutions involving geometry: A step towards efficient shape optimization. Comput. Methods Appl. Mech. Eng., 2014; 268:178-193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10-dimensional space, but the PGD solution provides a fairly accurate approximation (error 1%) using less that 150 terms in the PGD expansion. Copyright (c) 2015John Wiley & Sons, Ltd.]]>