arXiv subjects

Jens Visbech

Publications and source records attributed to Jens Visbech.

8 recordsLinked to original sources

arXiv2026

A high-order polynomial-corrected shifted boundary method for simulating fully nonlinear water waves

We present a novel unfitted computational framework for simulating fully nonlinear potential flow-based water waves. Focusing on wave propagation, we describe the core methodology, which involves a high-order polynomial-corrected shifted-boundary approximation on unfitted spectral elements. This approach allows for the simulation of a curved, highly time-dependent (moving and deforming) free surface affected by bathymetric changes. All that on a simple Cartesian background mesh without re-meshing or further approximations of non-affine geometric features. In addition, we highlight the importance of proper gradient recovery using a polynomial-preserving technique to accurately capture the vertical free-surface velocity. Ultimately, the goal is to develop a high-order convergent numerical scheme capable of simulating highly nonlinear waves over long periods. This is achieved through an arbitrary-order finite-difference approximation of the free surface combined with added hyperviscosity for numerical stability. We present verification and validation test cases for wave propagation in both periodic and finite domains. Emphasis is placed on convergence studies, the justification for using high-order approximations, the importance of optimal gradient recovery, long-time simulation of highly nonlinear waves, and nonlinear wave interactions with both affine and curved bathymetry changes, as well as with vertical walls, for highly nonlinear stream function waves and high-amplitude solitons. The novel computational model offers a comprehensive, all-in-one framework for simulating ocean waves and their interactions with offshore structures. It provides significant geometric flexibility, enabling boundaries such as the free surface to move and deform over time without the need to re-mesh a boundary-fitted mesh.

physics.flu-dyn↗

arXiv2026

Generalized high-order minimization-based polynomial corrections on unfitted spectral elements for the Poisson problem

Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.

math.NA↗

arXiv2026

Estimating Hydrodynamic Coefficients for Floating Offshore Structures from Movement Data Using Physics-Informed Neural Networks

We present a method for estimating the hydrodynamic coefficients in the Cummins equations using time-series data from a moving body, such as a floating offshore structure. The proposed data-driven method is based on incorporating the Cummins equations governing the dynamics of a structural body interacting with water waves into a physics-informed neural network (PINN), along with available motion data. The proposed method first estimates the structure's state in terms of translational and rotational degrees of freedom, and then solves the inverse problem to determine the hydrodynamic forces acting on the body, expressed in terms of added mass, damping coefficients, and/or hydrostatic restoring. The Cummins equations are formulated as a first-order system, and both state and parameter estimation are performed using PINNs. The method is verified on the free decay of a sphere and a box. The results demonstrate that it is possible to estimate the state and hydrodynamic coefficients accurately, although accuracy depends on the volume and quality of the movement data.

physics.flu-dyn↗

arXiv2025

A High-Order Spectral Element Solver for Steady-State Free Surface Flows

We present a spectral element solver for the steady incompressible Navier-Stokes equations subject to a free surface. Utilizing the kinematic behaviour of the free surface boundary, an iterative pseudo-time procedure is proposed to determine the a priori unknown free surface profile. The numerical model is implemented in the open-source finite element framework Firedrake, which enables the use of a high-order polynomial basis on unstructured meshes through weak formulations. Additionally, the curvature of the free surface and submerged bodies is incorporated through curvilinear elements obtained via transfinite linear blending, which conserves the high-order convergent properties of the overall scheme. The model is applied to several benchmark cases in two spatial dimensions. Initially, it addresses fixed-domain problems, including the lid-driven cavity flow and flows around bodies such as a cylinder and a NACA airfoil. Subsequently, with the presence of a free surface, it is extended to determine the flow around a bathymetry bump and a submerged NACA airfoil. The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.

math.NA↗

arXiv2025

FNPF-SEM: A parallel spectral element model in Firedrake for fully nonlinear water wave simulations

We present a new parallel spectral element solver, FNPF-SEM, for simulating linear and fully nonlinear potential flow-based water waves and their interaction with offshore structures. The tool is designed as a general-purpose wave model for offshore engineering applications. Built within the open-source framework Firedrake, the new FNPF-SEM model is designed as a computational tool capable of capturing both linear and nonlinear wave phenomena with high accuracy and efficiency, with support for high-order (spectral) finite elements. Additionally, Firedrake provides native support for MPI-based parallelism, allowing for efficient multi-CPU distributed computations needed for large-scale simulations. We demonstrate the capabilities of the high-order spectral element model through h- and p-convergence studies, and weak and strong scaling tests. Validation is performed against analytical solutions and experimental data for several benchmark cases, including nonlinear high-order harmonic generation and linear and nonlinear wave interactions with a cylinder and a breakwater. The new FNPF-SEM model offers a numerical framework for simulating wave propagation and wave-structure interactions, with the following key features: i) the ability to represent complex geometries through flexible, unstructured finite element meshes; ii) reduced numerical diffusion and dispersion by using high-order polynomial expansions; and iii) scalability to full- and large-scale simulations over long time periods through a parallel implementation.

math.NA↗

arXiv2023

A spectral element solution of the Poisson equation with shifted boundary polynomial corrections: influence of the surrogate to true boundary mapping and an asymptotically preserving Robin formulation

We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

math.NA↗

arXiv2023

Solving the complete pseudo-impulsive radiation and diffraction problem using a spectral element method

This paper presents a novel, efficient, high-order accurate, and stable spectral element-based model for computing the complete three-dimensional linear radiation and diffraction problem for floating offshore structures. We present a solution to a pseudo-impulsive formulation in the time domain, where the frequency-dependent quantities, such as added mass, radiation damping, and wave excitation force for arbitrary heading angle, $β$, are evaluated using Fourier transforms from the tailored time-domain responses. The spatial domain is tessellated by an unstructured high-order hybrid configured mesh and represented by piece-wise polynomial basis functions in the spectral element space. Fourth-order accurate time integration is employed through an explicit four-stage Runge-Kutta method and complemented by fourth-order finite difference approximations for time differentiation. To reduce the computational burden, the model can make use of symmetry boundaries in the domain representation. The key piece of the numerical model -- the discrete Laplace solver -- is validated through $p$- and $h$-convergence studies. Moreover, to highlight the capabilities of the proposed model, we present prof-of-concept examples of simple floating bodies (a sphere and a box). Lastly, a much more involved case is performed of an oscillating water column, including generalized modes resembling the piston motion and wave sloshing effects inside the wave energy converter chamber. In this case, the spectral element model trivially computes the infinite-frequency added mass, which is a singular problem for conventional boundary element type solvers.

math.NA↗

arXiv2022

A spectral element solution of the 2D linearized potential flow radiation problem

We present a scalable 2D Galerkin spectral element method solution to the linearized potential flow radiation problem for wave induced forcing of a floating offshore structure. The pseudo-impulsive formulation of the problem is solved in the time-domain using a Gaussian displacement signal tailored to the discrete resolution. The added mass and damping coefficients are then obtained via Fourier transformation. The spectral element method is used to discretize the spatial fluid domain, whereas the classical explicit 4-stage 4th order Runge-Kutta scheme is employed for the temporal integration. Spectral convergence of the proposed model is established for both affine and curvilinear elements, and the computational effort is shown to scale with $\mathcal{O}(N^p)$, with $N$ begin the total number of grid points and $p \approx 1$. Temporal stability properties, caused by the spatial resolution, are considered to ensure a stable model. The solver is used to compute the hydrodynamic coefficients for several floating bodies and compare against known public benchmark results. The results are showing excellent agreement, ultimately validating the solver and emphasizing the geometrical flexibility and high accuracy and efficiency of the proposed solver strategy. Lastly, an extensive investigation of non-resolved energy from the pseudo-impulse is carried out to characterise the induced spurious oscillations of the free surface quantities leading to a verification of a proposal on how to efficiently and accurately calculate added mass and damping coefficients in pseudo-impulsive solvers.

math.NA↗