ABSTRACT. We present two different numerical-wavetank models, driven by piston- and waveflap-wavemakers, that are respectively used to simulate shallow- and deep-water dynamics. The mathematical models are based on the variational principles (VP), for potential-flow equations, whose spatial and temporal discretizations are performed using respectively finite elements (FEM) and geometric time integrators, e.g. Störmer-Verlet (SV) and modified-midpoint (MMP) schemes, which are chosen because of their energy-conservation properties.

An extensive numerical validation is undertaken that involves a series of test cases for simulating regular and irregular waves generated by different parameters and characteristics in the numerical wavetank. Results are then compared against experimental data. These numerical-wavetank models are developed by implementing the potential-flow equations in the finite-element-based environment Firedrake. Both wavetanks have moving free-surface and wavemaker boundaries that must first be transformed into a stationary computational domain. These transformations are straightforward for the piston-wavemaker case because the motion is only time-dependent. However, for the waveflap case, the spatiotemporal nature of the waveflap motion complicates the transformation process. Hence, for the waveflap case, we have derived a novel two-step transformation that maps the moving domain onto a fixed one. The vertical (z-coordinate) is spatially discretized by using high-order Lagrange polynomials over one element while the horizontal (x- and y-coordinates) use first-order continuous Galerkin Lagrange polynomials over a large number of linear quadrilateral finite elements.

The numerical validation utilises a time-discrete VP of time-discrete weak formulations that are generated automatically within Firedrake's finite-element compiler architecture, thereby reducing both the likelihood of human error and the time taken to both develop and implement the code. The numerical simulations were undertaken with different sets of parameters which, being difficult to optimise, have prevented us from reducing significantly the computational times required: guidance on how to circumvent this limitation will be sought from the Firedrake team.

ABSTRACT. In this talk we consider the Stokes–Onsager–Stefan–Maxwell (SOSM) equations, which model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. The equations account for both the diffusive interactions between chemical species and the bulk convection. Our aim is to develop computationally efficient high-order finite element schemes that discretize these nonlinear equations in two and three spatial dimensions. Because the SOSM equations relate many unknown variables (e.g. the bulk and species velocities, pressure, concentrations, chemical potentials, etc.), this is a difficult task. In particular, there are many choices of which variables should be explicitly solved for in the formulation, and it is not clear which discrete finite element function spaces should be employed. To tackle this challenge, we derive a novel weak formulation of the SOSM problem in which the species mass fluxes are treated as unknowns. We show that this new formulation naturally leads to a large class of high-order finite element discretizations that are straightforward to implement and have desirable linear-algebraic properties. Moreover, from a theoretical standpoint, we are able to prove that when applied to a linearized version of the SOSM problem, the proposed finite element schemes are convergent. Our findings are illustrated with numerical experiments.

ABSTRACT. The formation of extreme waves arising from the interactions of two and three line-solitons (respectively denoted by cases SP2 and SP3) with equal far-field amplitudes is investigated numerically using a water-wave model based on potential-flow equations (PFE), following the promising and inspiring results presented in our previous work [Choi, J., Bokhove, O., Kalogirou, A., Kelmanson, M., Numerical Experiments on Extreme Waves Through Oblique-Soliton Interactions, Water Waves 4, 139–179 (2022)] on a similar scenario using the Benney-Luke equation.

When building the computational model using Firedrake, two approaches are adopted, the first of which follows a more traditional procedure in which the weak formulations are derived manually from a transformed variational principle (VP) and formulated explicitly in the code. By contrast, the second approach is based on the time-discretised VP, for which the weak formulations are generated automatically and implemented implicitly using Firedrake’s inherent ‘derivative’ functionality; this approach both shortens the development time considerably and reduces the risk of introduction of human error. It is discovered that the maximum amplifications obtained from the two approaches are generally consistent with each other.

Motivated by the question as to what extent the interacted line-solitons would endure and attain the theoretical maximum amplitudes determined from the Kadomtsev-Petviashvili equation (KPE) on a more realistic PFE model, we employ consistent variational discretisations in space and time to ensure the conservation of physical properties robustly. Second- and fourth-order continuous Galerkin polynomials, respectively CG2 and CG4, are used in the horizontal layers while, in the vertical direction, spectrally accurate Gauss-Lobatto-Legendre (GLL) polynomials are deployed.

For both SP2 and SP3 simulations, corresponding analytical web-soliton solutions of the KPE on an infinite horizontal plane are adapted to seed initial conditions on an x-periodic (truncated yet sufficiently large) domain of the PFE system. For one spatial resolution in each of SP2 and SP3, a series of simulations with repeatedly halved time steps are undertaken to verify the order of temporal convergence. For SP2, it is found that the well-known theoretical fourfold amplification of KPE can be reached and approximately sustained in the present model, while, in the case of SP3, due to stronger nonlinear effects there is comparatively less consistency between the numerically achieved amplification factor and the newly theoretically established KPE maximum ninefold amplification.

ABSTRACT. We present a fluid-structure-interaction formulation for elongated micro-organisms, described as 1D entities immersed in viscous fluids. Challenging aspects of modeling such one-dimensional bodies in the soft-biomatter realm are the large deformations usually involved and the complex response of the fluid for their propulsion and guidance. The proposed formulation has been recently published [1] and allows to simulate 1D microswimmers with finite thickness, such as Kirchhoff rods, experiencing very large displacements and deformations in generalized Newtonian fluids (e.g. Carreau-Yasuda). In this presentation, we extend the results shown in [1] by first discussing some error estimates of a linearized version of the problem. Second, interaction of the swimmer with a concentration field is also studied. Third, a reformulation of the Fluid-Structure-Interaction algorithm using Nitsche’s method to impose the kinematic constraints, is proposed so as to enable for slip boundary conditions. Finally, some ongoing work on possible control strategies of the swimmer are discussed.

[1] R.F. Ausas, C.G. Gebhardt and G.C. Buscaglia, A finite element method for simulating soft active non-shearable rods immersed in generalized Newtonian fluids. Communications in Nonlinear Science and Numerical Simulation, Vol. 108, pp. 106213, 2022.

ABSTRACT. We present multigrid solvers for the high-order FEM de Rham complex with the same time and space complexity as sum-factorized operator application. Our approach relies on new finite elements with orthogonality properties on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions, in the separable case. In the non-separable case, the method can be applied to a sparse auxiliary operator.

We illustrate our preconditioning approach by solving mixed formulations of the Hodge–Laplacian and the vorticity-velocity-pressure formulation of Stokes flow, for which we observe robustness with respect to the mesh size and the polynomial degree.