Marx Chhay

Geometric numerical schemes for the KdV equation

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.

Proper generalized decomposition for solving coupled heat and moisture transfer

This paper proposes a reduced order model to simulate heat and moisture behaviour of material based on proper general decomposition (PGD). This innovative method is an a priori model reduction method. It proposes an alternative way for computing solutions of the problem by considering a separated representation of the solution. PGD offers an interesting reduction of numerical cost. In this paper, the PGD solution is first compared with a finite element solution and the commercial validated model Delphin in an 1D case. The results show that the PGD resolution techniques enable the field of interest to be represented with accuracy, with a relative error rate of less than 0.1%. The study remains in the hygroscopic range of the material. As the numerical gain of the method becomes interesting when the space dimension increases, this resolution strategy was then used on a 2D multi-layered test case. The dynamics and amplitude of hygrothermal fields are perfectly represented by the PGD solution. Temperature and vapour pressure modelled with PGD can be used for post-processing and analysing the behaviour of an assembly.

Estimation of temperature-dependent thermal conductivity using proper generalised decomposition for building energy management

A proper generalised decomposition for solving inverse heat conduction problems is proposed in this article as an innovative method offering important numerical savings. It is based on the solution of a parametric problem, considering the unknown parameter as a coordinate of the problem. Then, considering this solution, all sets of cost function can be computed as a function of the unknown parameter of the defined domain, identifying the argument that minimises the cost function. In order to illustrate the applicability, the method is used to solve a non-linear inverse heat conduction problem to determine a temperature-dependent thermal conductivity. Then, a comparison is carried out with the local sensitivity and the genetic algorithm methods. It is shown that the proper generalised decomposition method estimates the unknown parameter with the same accuracy as the other two methods. Due to its advantage in terms of reducing the complexity, the method was then used to solve a transient three-dimensional non-linear heat transfer inverse problem. The results have shown that the method is appropriate to determine the unknown parameter with a low computational cost. Furthermore, the main advantage of the technique is its low capacity for storage, which can be used, as an inverse method, for building energy management and extended to evaluate thermal bridges from on-site measurements.

Numerical study of the generalised Klein–Gordon equations

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets.