Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Efficient p-multigrid spectral element model for water waves and marine offshore structures

In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave-body interaction with high computational efficiency within a single modellingapproach. We design and propose an efficientO(n)-scalable computational procedure based on geometric p-multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high-order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p-multigrid spectralelement model can take advantage of the high-order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h-multigrid approaches. We provide numerical benchmarks for the algorithmic and numerical efficiency of the iterative geometric p-multigrid solver. Results of numerical experiments are presented for wave propagation and for wave-body interaction in an advanced case for focusing design waves interacting with a FPSO. Our study shows, that the use of iterative geometric p-multigrid methods for theLaplace problem can significantly improve run-time efficiency of FNPF simulators.Comment: Submitted to an international journal for peer revie

A high-order and mesh-free computational model for non-linear water waves

In this paper, we present the ongoing developments of a novel computational model for non-linear water waves that aims to provide a suitable framework for wave-structure inter- action. The proposed model is based on radial basis function-generated finite differences, which allow for arbitrary and moving boundaries without the use of ghost nodes. In order to take advantage of the mesh-free setting, we propose a node generation strategy, suitable for moving boundaries. Numerical properties of the proposed model are investigated and finally the model is benchmarked. The proposed model is expected to provide a suitable computational framework for wave-structure interaction problems, due to its geometric flexibility and high-order nature

Comparative Numerical Study on Focusing Wave Interaction with FPSO-like Structure

Evaluating the interactions between offshore structures and extreme waves plays an essential role for securing the surviv-ability of the structures. For this purpose, various numerical tools—for example, the fully nonlinear potential theory (FNPT),the Navier–Stokes (NS) models, and hybrid approaches combining different numerical models—have been developed andemployed. However, there is still great uncertainty over the required level of model fidelity when being applied to a widerange of wave-structure interaction problems. This paper aims to shed some light on this issue with a specific focus on theoverall error sourced from wave generation/absorbing techniques and resolving the viscous and turbulent effects, by com-paring the performances of three different models, including the quasi-arbitrary Lagrangian Eulerian finite element method(QALE-FEM) based on the FNPT, an in-house two-phase NS model with large-eddy simulation and a hybrid model couplingthe QALE-FEM with the OpenFOAM/InterDymFoam, in the cases with a fixed FPSO-like structure under extreme focus-ing waves. The relative errors of numerical models are defined against the experimental data, which are released after thenumerical works have been completed (i.e., a blind test), in terms of the pressure and wave elevations. This paper providesa practical reference for not only choosing an appropriate model in practices but also on developing/optimizing numericaltools for more reliable and robust predications

A Comparative Study on the Nonlinear Interaction Between a Focusing Wave and Cylinder Using State-of-the-art Solvers: Part A

This paper presents ISOPE’s 2020 comparative study on the interaction between focused waves and a fixed cylinder. The paper discusses the qualitative and quantitative comparisons between 20 different numerical solvers from various universities across the world for a fixed cylinder. The moving cylinder cases are reported in a companion paper as part B (Agarwal, Saincher, et al., 2021). The numerical solvers presented in this paper are the recent state of the art in the field, mostly developed in-house by various academic institutes. The majority of the participants used hybrid modeling (i.e., a combination of potential flow and Navier–Stokes solvers). The qualitative comparisons based on the wave probe and pressure probe time histories and spectral components between laminar, turbulent, and potential flow solvers are presented in this paper. Furthermore, the quantitative error analyses based on the overall relative error in peak and phase shifts in the wave probe and pressure probe of all the 20 different solvers are reported. The quantitative errors with respect to different spectral component energy levels (i.e., in primary, sub-, and superharmonic regions) capturing capability are reported. Thus, the paper discusses the maximum, minimum, and median relative errors present in recent solvers as regards application to industrial problems rather than attempting to find the best solver. Furthermore, recommendations are drawn based on the analysis

A high-order spectral element unified boussinesq model for floating point absorbers

International audienceNonlinear wave-body problems are important in renewable energy, especially in case of wave energy converters operating in the near-shore region. In this paper we simulate nonlinear interaction between waves and truncated bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results-if applied within the underlying assumptions of dispersion and nonlinearity-but with a significant reduction in computational effort

Comparative numerical study on focusing wave interaction with FPSO-like structure

Evaluating the interactions between offshore structures and extreme waves plays an essential role for securing the survivability of the structures. For this purpose, various numerical tools—for example, the fully nonlinear potential theory (FNPT), the Navier–Stokes (NS) models, and hybrid approaches combining different numerical models—have been developed and employed. However, there is still great uncertainty over the required level of model fidelity when being applied to a wide range of wave-structure interaction problems. This paper aims to shed some light on this issue with a specific focus on the overall error sourced from wave generation/absorbing techniques and resolving the viscous and turbulent effects, by comparing the performances of three different models, including the quasi-arbitrary Lagrangian Eulerian finite element method (QALE-FEM) based on the FNPT, an in-house two-phase NS model with large-eddy simulation and a hybrid model coupling the QALE-FEM with the OpenFOAM/InterDymFoam, in the cases with a fixed FPSO-like structure under extreme focusing waves. The relative errors of numerical models are defined against the experimental data, which are released after the numerical works have been completed (i.e., a blind test), in terms of the pressure and wave elevations. This paper provides a practical reference for not only choosing an appropriate model in practices but also on developing/optimizing numerical tools for more reliable and robust predications