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A discontinuous finite element method for the hydroelastic analysis of submerged structures

Zachary J. Wegert, Oriol Colomés, Michael H. Meylan, Ngamta Thamwattana, Vivien J. Challis

arXiv:2609.08313v2math.NAcs.NA

TL;DR

Submerged plate wave-energy converters motivate a dedicated FEM approach, as direct FEM studies are relatively few. The method combines fluid and plate Galerkin formulations with non-local radiation boundaries; validation agrees excellently with a hypersingular boundary integral method and three-dimensional cases satisfy energy balance.

  • Problem

    Direct FEM studies of submerged plate WECs are relatively uncommon, despite FEM’s flexibility for geometries and physical state equations relevant to thin elastic plates.

    Source: §1 · p. 2

  • Method

    The method couples a symmetric DG fluid formulation, non-local radiation boundary operators, and a continuous/discontinuous Galerkin plate formulation.

    Source: §6 · p. 37

  • Results

    The method showed excellent agreement with a hypersingular boundary integral method for a horizontal plate, and three-dimensional reflection and transmission coefficients satisfied energy balance.

    Source: §6 · p. 38

  • Takeaways & Limitations

    The formulation is relevant to submerged plate WEC modelling and is described as extendable to fixed or rigid structures and variable bottom topography.

    Source: §1 · p. 3, §6 · p. 38

    The present work neglects piezoelectric effects; extending the formulation to piezoelectric wave energy converters is future work.

    Source: §6 · p. 38

Abstract

from arXiv · show

This paper develops a finite element method for wave-structure interaction problems arising in the hydroelastic modelling of submerged elastic plates. The approach is formulated for both two-dimensional settings and three-dimensional channels. For the fluid part of the problem, we use a symmetric discontinuous Galerkin scheme that captures the discontinuity in the potential across the plate. To impose the Sommerfeld radiation conditions, we take inspiration from the Dirichlet-to-Neumann map approach and use the analytic solution in the semi-infinite domains to construct appropriate non-local boundary conditions. For the plate, we use a continuous/discontinuous Galerkin method to resolve the 4$^{\rm th}$ order operator without using continuously differentiable finite elements. We prove well-posedness of the method, showing that the resulting sesquilinear form is bounded and satisfies the G{\aa}rding inequality, leading to a unique and stable solution. We implement our formulation as an open-source tool using the Gridap package in Julia. We validate our method against a hypersingular boundary integral method, finding excellent agreement. The flexibility of the methodology is demonstrated through simulations of a two-dimensional slanted plate and three-dimensional plate geometries including centred rectangular plates, offset rectangular plates, and annulus-shaped plates. The finite element method presented here can be readily applied to model wave energy conversion of piezoelectric bimorphs. Furthermore, the method could easily be extended to consider wave scattering from fixed or rigid structures or from variable bottom topography, and is not limited to hydroelastic applications.

1. Introduction

The paper develops a dedicated finite element framework for submerged plate wave-energy converters, addressing the relative scarcity of direct FEM studies in this area. It builds on hydroelastic FEM work for floating structures and targets the submerged-plate setting.

1. Introduction

  • FEM offers flexibility for physical state equations and geometries, making it well-suited to thin elastic plates and wave-energy modelling.The paper notes that FEM is less extensively developed for water-wave scattering than standard Green’s-function and boundary-integral approaches.Source: §1 · pp. 1–2
  • Submerged plates are motivated for WECs because they experience lower wave loads, smoother water motion, and improved durability than surface plates.The introduction links these differences to energy conversion efficiency, structural stress, and maintenance.Source: §1 · p. 2
  • Submerged plate WEC research has used FVM, BEM, and SPH more often than direct FEM, which remains relatively uncommon.Many existing FEM studies rely on commercial packages or coupled CFD–FEM approaches.Source: §1 · p. 2
  • The paper develops a dedicated FEM framework for submerged plates, extending prior monolithic FEM work on floating structures to a submerged-plate problem.The formulation is relevant to submerged plate piezoelectric WECs, while neglecting piezoelectric effects in this work.Source: §1 · pp. 2–3
  • The framework is proposed as a basis for extensions to non-uniform ocean depth, non-homogeneous structures, and non-linear bending.The paper also positions it as directly relevant to submerged plate piezoelectric WEC modelling.Source: §1 · p. 3

2. Problem setup

The model describes submerged elastic plates interacting with water waves in two-dimensional fluid and three-dimensional channel settings. It uses frequency-domain assumptions and allows free, clamped, or simply-supported plate boundary conditions.

2. Problem setup

  • This section describes the model and assumptions and explains the imposition of radiation conditions in finite domains, supplying setup context for the paper’s finite-domain treatment.Source: §2 · p. 3

2.1. Hydroelastic model

  • The model neglects piezoelectric effects, though the formulation is described as extendable to piezoelectric WEC modelling using complex-valued rigidity coefficients.That extension is identified as future work.Source: §2.1 · p. 3

2.1.1. Two-dimensional model

  • In two dimensions, a one-dimensional plate bends in an inviscid, incompressible, irrotational fluid under time-harmonic motion in the frequency domain.The plate displacement may satisfy free, clamped, or simply-supported boundary conditions.Source: §2.1.1 · pp. 3–4

2.1.2. Three-dimensional channel model

  • The three-dimensional channel model treats the submerged plate as a thin Poisson–Kirchhoff plate, with a symmetric order-4 rigidity tensor in its plate equation.The setup includes a plane incident wave as an example, permits non-plane incident waves, and imposes slip conditions on channel sides.Source: §2.1.2 · pp. 5–6, Figure 2 · p. 5

3. Finite element formulation

The formulation combines discontinuous Galerkin treatment of the fluid with a continuous/discontinuous Galerkin treatment of the plate. Non-local boundary operators impose radiation conditions in finite domains, and the discrete problem couples the fluid and plate forms.

3.1. Notation

  • The notation partitions the fluid and plate into elements and defines skeletons, jumps, means, and broken function spaces for the variational formulation.Across the plate, jump values are ordered using the fluid values below and above it.Source: §3.1 · p. 6

3.2. Fluid

  • The fluid weak form adds symmetric interior-penalty and penalty terms to enforce continuity of the potential while retaining the jump across the plate.These additions are consistent when the potential solves the governing equations.Source: §3.2 · p. 7

3.2.1. Imposing Sommerfeld radiation conditions in two dimensions

  • For two-dimensional radiation conditions, analytic solutions in the semi-infinite exterior domains produce a non-local boundary operator at the finite-domain ends.The eigenfunction sums are truncated, and the resulting boundary contribution acts on degrees of freedom at the inlet and outlet.Source: §3.2.1 · pp. 7–8

3.2.2. Imposing Sommerfeld radiation conditions for the three-dimensional channel

  • For the three-dimensional channel, the radiation treatment uses eigen-expansion coefficients and truncated modal sums to form the boundary operators.The construction takes normal derivatives at the inlet and outlet boundaries.Source: §3.2.2 · pp. 9–10

3.3. Plate

  • The plate’s fourth-order operator is handled with a continuous/discontinuous Galerkin formulation, avoiding C1 finite elements.The derivation adds symmetric DG and penalty terms; the resulting plate form includes bending, inertia, and fluid-coupling contributions.Source: §3.3 · p. 10
  • Free and simply-supported plate conditions share one plate operator, while clamped conditions weakly enforce zero normal displacement gradient using symmetric interior penalty.The boundary contributions depend on the selected plate boundary condition.Source: §3.3 · pp. 10–11

3.4. Discrete formulation

  • The discrete formulation seeks fluid-potential and plate-displacement functions whose fluid and scaled plate bilinear forms equal the load for all discrete test functions.The scaling parameter is introduced for stability and dimensional consistency, and the plate form depends on the boundary condition.Source: §3.4 · p. 11

4. Numerical analysis

Section 4 analyzes the discrete formulation, establishing continuity properties and discrete stability bounds under the stated assumptions. These results yield conditional well-posedness, unique stable solutions, and quasi-optimal error control.

4.1. Preliminary arguments

  • The analysis assumes bounded uniformly elliptic bending rigidity, fixed finite orthogonal radiation modes with outgoing or decaying branches, and continuous non-resonance.Source: §4.1 · pp. 13–14
  • The analysis requires a compatible shape-regular split into Lipschitz fluid subdomains; free interior plate tips violate it and invalidate the regularity-based quasi-optimal estimate, but not stability results.Source: §4.1 · p. 13
  • If Assumption 3 holds, there exists a subsequence converging to ϕ ∈ H1(Ω\Γ), with no continuity required across Γ, and w ∈ H2(Γ); moreover, ϕh and ∇wh converge strongly in L2(Ω) and L2(Γ), respectively.Source: §4.1 · p. 15

4.2. Discrete continuity

  • The coupled sesquilinear form is continuous in the discrete norm, with a mesh-independent bound.The proof bounds the fluid, coupling, plate, and skeleton terms.Source: §4.2 · pp. 15, 20
  • Assembly of the independent contributions yields a global discrete-continuity bound with a mesh-independent constant.The bound is assembled using the triangle inequality and a maximum of the individual contribution constants.Source: §4.2 · p. 20

4.3. Discrete Gårding inequality

  • The coupled sesquilinear form satisfies a discrete Gårding inequality on Xh for µp = 1/ρw, subject to the stated recovery-operator mesh-shape-regularity condition.The proof bounds the radiation, fluid-skeleton, Robin/coupling boundary, and structure contributions; the outgoing radiation term is nonnegative.Source: §4.3 · pp. 20, 21, 25

4.4. Extension to the exact solution

  • Extended continuity holds for exact solutions with broken regularity in an augmented norm, with a constant independent of mesh size.This extension handles mean-flux and mean-moment terms that require separate bounds when the exact solution is not piecewise polynomial.Source: §4.4 · pp. 25–26

4.5. Well-posedness

  • Under Assumption 4, the exact solution is consistent with the discrete formulation: B(x, yh) = L(yh) for every yh in Xh.The identity follows by blockwise integration by parts and the assumed interface and boundary conditions.Source: §4.5 · pp. 27–28
  • Any loss of uniqueness in the homogeneous continuous problem can occur only at isolated interior resonance frequencies.A nontrivial solution carries no radiated energy, and the stated condition for discrete well-posedness assumes the resulting interior resonance condition is satisfied.Source: §4.5 · p. 28
  • For sufficiently small h and under Assumptions 1, 2, and 5, the discrete problem satisfies an inf-sup condition and has a unique, stable solution.Under Assumption 4, the Galerkin solution also satisfies a quasi-optimal error bound.Source: §4.5 · p. 29
  • For sufficiently small h, the discrete inf-sup bound ensures a unique solution for every L and yields stability.The bound holds for h < h0 with γ > 0.Source: §4.5 · p. 31
  • Under Assumption 4, Galerkin orthogonality yields quasi-optimal error control; an unconditional bound also follows without Assumption 4.Source: §4.5 · p. 31

5. Numerical results

The numerical examples validate the finite element method against Wegert et al. and demonstrate its application to slanted and three-dimensional submerged plates. Changing plate geometry requires mesh generation but no other code adjustments.

5.1. Implementation

  • The open-source implementation uses Gridap in Julia, Gmsh for meshes, and second-order Lagrange polynomials with mesh-scaled penalty parameters.The Sommerfeld radiation operators are discretised using outer products; the polynomial order can be increased.Source: §5.1 · p. 31

5.2. Validation

  • The proposed finite element scheme shows excellent displacement agreement with Wegert et al. across mesh sizes, even extremely coarse ones; reflection and transmission results agree well over periods 1 s to 5 s at approximate hsize = 0.1 m.Source: §5.2 · p. 32

5.3. Slanted plates

  • For slanted plates, extending the implementation requires only mesh-generation changes, unlike the modifications required by Wegert et al..The example compares free and simply-supported ends, with |R| = 0.756 and |T| = 0.655 for free ends, versus |R| = 0.969 and |T| = 0.248 for simply-supported ends.Source: §5.3 · p. 32, Figure 6 · p. 34

5.4. Three-dimensional channel

  • Three-dimensional examples cover centred and offset rectangular plates and an annulus; the symmetric mode n = 2 coefficients are large for the offset plate and small for the annulus.The paper attributes these differences to system non-symmetry and symmetry, respectively.Source: §5.4 · pp. 35–37
  • The three-dimensional examples use an iterative induced dimension reduction solver with incomplete LU preconditioning, which is sufficient for these cases but is not scalable.The paper identifies scalable linear solvers as future work.Source: §5.4 · p. 35
  • For the three-dimensional examples, the non-local radiation operator uses M = 5 and N = 5 modes, with N chosen to capture propagating modes.The paper states that N should be at least the number of propagating modes.Source: §5.4 · p. 35
  • The computed energy-balance residuals are |ξ| = 3.09 × 10^-9 for the centred plate, 5.63 × 10^-10 for the offset plate, and 1.12 × 10^-8 for the annulus.The paper reports that these solutions approximately satisfy energy conservation.Source: §5.4 · p. 37

6. Conclusions

The method is proved well-posed and stable, validated against a boundary integral method, and applied to two- and three-dimensional submerged-plate cases. The authors also describe its implementation and identify extensions for future work.

6. Conclusions

  • A symmetric discontinuous Galerkin fluid scheme and a continuous/discontinuous Galerkin plate method yield a bounded sesquilinear form, a discrete Gårding inequality, and a unique, stable solution.The formulation captures the potential discontinuity across the plate and resolves the fourth-order plate operator without continuously differentiable finite elements.Source: §6 · p. 37
  • The formulation is implemented with Gridap in Julia, and the freely available code is intended to support verification and numerical approximation of more complicated problems.Gridap's programming interface keeps the code near one-to-one with the mathematical notation and permits a range of cases with minimal changes.Source: §6 · pp. 37–38
  • The method agrees excellently with a hypersingular boundary integral method, and the three-dimensional propagating-wave reflection and transmission coefficients satisfy energy balance.The comparison is for a horizontal submerged plate; the three-dimensional cases include centred and offset rectangles and an annulus.Source: §6 · p. 38
  • The method covers two-dimensional slanted plates and three-dimensional centred, offset, and annulus-shaped plates; only mesh generation needed changing for the slanted case.For the annulus case, Figure 10 visualizes real and imaginary free-surface and plate displacements, and Table 4 reports propagating-mode reflection and transmission coefficients.Source: §6 · p. 38, Figure 10 · p. 38, Table 4 · p. 38
  • The work focuses on hydroelastic problems; the authors identify radially infinite oceans, piezoelectric converters, and unfitted finite elements for plate-shape optimization as future extensions.They also describe applications to fixed or rigid structures and variable bottom topography as straightforward extensions that simplify the problem.Source: §6 · p. 38

CRediT authorship contribution statement

The contribution statement assigns writing, conceptualization, and project leadership across the five authors, with technical and software responsibilities distributed among them.

CRediT authorship contribution statement

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