Principal Investigator: Stéphane Clain
Funder: Portuguese Foundation for Science and Technology (FCT)
Beneficiaries: University of Minho (UMinho), University of Coimbra (UCoimbra), Association for the Research and Development of Sciences (FCiências.ID)
Call: Call for R&D Projects in All Scientific Domains
Period: 14/12/2018 – 13/12/2022 (4 years)
Reference: PTDC/MAT-APL/28118/2017, LISBOA-01-0145-FEDER-028118, POCI-01-0145-FEDER-028118
Abstract and objectives:
The first part of the project addresses the efficiency of numerical methods, focusing on:
Increasing the spatial order of convergence, enabling the same accuracy with significantly fewer unknowns. A major limitation arises with curved boundaries, where domain approximations reduce any numerical method to second-order accuracy. Finite element (FEM) and discontinuous Galerkin (DG) methods employ isoparametric elements, which require local transformations, additional computational effort, and complex curved meshing algorithms. A novel technique, capable of preserving the optimal convergence order, has been successfully tested for 2D linear convection–diffusion problems in the finite volume method (FVM). We now aim to extend this approach to complex 3D geometries with a broad class of boundary conditions.
Designing very high-order implicit time schemes for unsteady equations. These schemes are not unconditionally stable, as the time step remains constrained by the smallest cell size. This typically requires prohibitively small time steps. Time–space discretisations provide a promising alternative, enabling unconditional schemes and relaxing stability restrictions. While a 2nd-order implicit formulation for Maxwell’s equations has been proposed in the literature, we aim to extend this strategy through time–space coupled discretisation.
Exploiting modern hardware architectures, where personal computers gain performance through vector units, many-core processors, and hierarchical cache systems. Efficient algorithms must be designed to leverage these capabilities. The alternating direction implicit (ADI) method decomposes 2D or 3D problems into a series of independent 1D problems solved iteratively. Although recent studies have developed a novel ADI paradigm, they do not yet fully exploit modern hardware. We propose a new ADI formulation specifically optimised for contemporary architectures, enabling many 1D problems to be solved concurrently with available threads, leading to much faster codes.
Developing novel approaches for fluid–structure interfaces and moving boundaries. In Eulerian formulations, interfaces cut through cells, leading to inaccuracies in their representation and spurious diffusion. Smoothed particle hydrodynamics (SPH) is well-suited for compressible and weakly compressible flows with moving interfaces. However, ensuring consistency remains challenging due to the so-called E0-error, despite various attempted corrections. We propose a novel SPH–FVM hybrid formulation, introducing a new convolution compatible with the space deformation induced by particle displacements. This ensures consistency, better approximating the physics of the problem.
The second part of the project concerns applications, namely:
Thermoplastic polymer manufacturing processes demand large amounts of energy for heating, melting, forming, and cooling along the production line. Numerical tools, typically based on 2nd-order FVM, are widely used to predict temperature evolution and guide experimental validation. However, such tools often require highly refined meshes, resulting in large computational costs. By developing more efficient and higher-order accurate methods, we aim to enable real-time simulations that can be integrated into the cooling process control loop, thereby increasing efficiency and reducing energy consumption.
Coastal structures in tsunami-prone regions must be designed to withstand extreme hydrodynamic forces while balancing safety requirements and construction costs. FVM is commonly adopted to simulate large-scale oceanic flows, while SPH methods are better suited for the highly nonlinear tsunami–structure interactions. However, SPH currently suffers from lack of consistency and strong numerical dissipation. The proposed development of a consistent SPH formulation will enable a reliable and efficient numerical tool for assessing tsunami impacts on coastal structures.
The retina, as the only visible part of the central nervous system, can be directly imaged using non-invasive optical devices. A common approach to assess the progression of diabetic macular edema (DME) and neurodegenerative diseases is to monitor retinal thickness via optical coherence tomography (OCT). While OCT distinguishes between healthy and DME patients, it cannot directly capture changes at the cellular scale. Thus, a rigorous study of how electromagnetic waves propagate through the retinal layers is required. Previously, we developed a 3D multilayer Monte Carlo model of the retina, where the optical properties of each layer were obtained from full-wave solutions of Maxwell’s equations using a DG-FEM formulation. Building on this, we now aim to develop an advanced OCT computational model based on time–space explicit DG-FEM methods, enabling the study of normal ageing and neurodegenerative diseases without relying solely on morphological alterations.