nonlinear wave-body interaction using a mixed ...

Proceedings of the 37th International Conference on Ocean, Offshore and Arctic Engineering OMAE 2018 June 17-22, 2018, Madrid, Spain

77692

NONLINEAR WAVE-BODY INTERACTION USING A MIXED-EULERIAN-LAGRANGIAN SPECTRAL ELEMENT MODEL

Carlos Monteserin Allan P. Engsig-Karup ∗ Department of Applied Mathematics Department of Applied Mathematics and Computer Science and Computer Science Technical University of Denmark Center for Energy Resources Engineering e-mail: [email protected] Technical University of Denmark Matematiktorvet, Building 303b, Room 108 DK-2800 Kgs. Lyngby (Denmark) e-mail: [email protected]

ABSTRACT We present recent progress on the development of a new fully nonlinear potential flow (FNPF) model for estimation of nonlinear wave-body interactions based on a stabilised unstructured spectral element method (SEM). We introduce new proofof-concepts for forced nonlinear wave-body interaction in two spatial dimensions to establish the methodology in the SEM setting utilising dynamically adapted unstructured meshes. The numerical method behind the proposed methodology is described in some detail and numerical experiments on the forced motion of (i) surface piercing and (ii) submerged bodies are presented.

are based on linear radiation-diffraction theory or strip theory. Lately the use of computational fluid dynamics (CFD) have been put forward for these cases [1] but CFD is very computational demanding for wave propagation. Models based on fully nonlinear potential flow equations (FNPF) are known to be more computationally expensive than the simplified models, not only compared to radiation-diffraction models for wave-body interaction problems, but also compared to nonlinear asymptotic wave models. Such numerical models are in widespread use for coastal engineering applications [2] and are typically used for dispersive and nonlinear wave propagation but needs extension to handle nonlinear wave-body interactions. However, compared to CFD that represents the most expensive computational approach to nonlinear wave-body interactions, the FNPF models are relatively cheap, and can thus be viewed as a compromise between accuracy and computational speed.

INTRODUCTION Accurate simulations of nonlinear wave-induced loads on structures, and the subsequent nonlinear wave-body interaction for moving bodies, are of key importance in applications related to ocean engineering and naval architecture. The nonlinear effects are known to be important in cases of more extreme loads and motions, and these are essential to account for in design of bodies or structures. Examples are e.g. design loads for ultimate limit strength (ULS) on marine vessels or other offshore structures, and design of wave energy devices working in resonance. The standard engineering tools for wave-body interaction

∗ Address

all correspondence to this author.

Claes Eskilsson Research Institutes of Sweden e-mail: [email protected] Department of Civil Engineering Aalborg University e-mail: [email protected]

The design of efficient and flexible numerical methods that can handle both the nonlinear wave propagation and wave-body interactions is non-trivial. Numerical models based on FNPF equations have traditionally been designed to focus on either wave propagation or wave-body interaction. FNPF models for wave propagation typically use high-order methods (e.g. highorder finite differences [3, 4] or spectral methods [5, 6]) in order to gain efficiency due to the lower degrees of freedom needed to accurately model wave propagation over many wavelengths. The 1

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z

use of σ -transformation [6, 7] further increases the efficiency but limits the use to only bottom-mounted structures. In contrast, FNPF models for wave-body interactions are generally not based on σ -transforms and often developed based on low-order methods (e.g. finite elements [7–9] or boundary elements [10]) in order to take the geometric flexibility into account. The finite element models uses the Mixed-Eularian-Lagrangian (MEL) [11] or similar [9] methods for capturing the free surface. The present work aims to combine the computational efficiency and scalability of using high-order methods with the geometrical flexibility of the finite element method by using a high-order finite element method, namely the spectral element method (SEM) due to Patera (1984) [12]. The framework for numerical approximations using the spectral element method have already proven useful in advanced hydrodynamics applications, see e.g. [13]. However, until recently the SEM model has with limited success been applied to free surface flows governed by the FNPF equations. The main issue seems to have been the general mesh instability problem as first reported in [14] for a SEM-MEL model and likewise for a FEM-MEL model [15] using triangular unstructured meshes. Lately new stabilised high-order SEM models for the FNPF equations have been presented: (i) a SEM-Eulerian type [16] targeted towards wave propagation and (ii) a SEM-MEL type [17] focused on wave interaction with fixed structures. In these recent works it has been shown how the use of vertically aligned finite elements forming a quadrilateral layer (in 2D) just below the free surface prevented any sign of mesh instability for all polynomial orders, even with gradient recovery based on global L2 Galerkin projections. This remedy has opened up for a flexible discretization of the fluid domain using hybrid meshes of unstructured (possibly curvilinear) triangles close to the body and curvilinear quadrilateral elements near the free surface and far from the body. Arbitrarily shaped bodies can be handled, and the spatial resolution is adjusted by controlling element sizes. In combination with high-order basis functions it is possible to balance the cost of discretization versus accuracy enabling a compromise between efficient numerical solution of wave-body applications with engineering accuracy. In the present work we present the extension in 2D of the stabilised SEM-MEL model [17] to account for moving bodies. We employ standard Galerkin projection of the FNPF equations to obtain a semi-discrete system of equations. This system is solved using a method of lines approach with an efficient explicit fourth-order Runge-Kutta (RK) method for the time-integration. Two boundary value problems are solved every step in simulation to obtain the velocity and acceleration potentials using the acceleration potential method [18]. This method is used to not only obtain the velocity field, but also to provide accurate approximation of wave-induced hydrodynamic forces on the body. The model is benchmarked for the analysis of the nonlinear flows produced by forced heave motion of (i) a surface-piercing trun-

⌘

z=⌘

FS

x body

⌦

h

b

z=

h

FIGURE 1: Notations for the fluid domain with a surface piercing

cylinder.

cated body [19] and (ii) a submerged cylinder in prescribed motion [20].

GOVERNING EQUATIONS The governing equations for FNPF equations is expressed in the form of a MEL formulation due to Longuet-Higgins & Cokelet (1976) [11]. Let the fluid domain Ω ∈ Rd (d = 2) be bounded, connected domain with piece-wise smooth boundary Γ ∈ Rd−1 . Let T : t ≥ 0 be the time domain. We seek a scalar velocity potential function φ (x, z,t) : Ω × T → R satisfying the Laplace problem φ = φ˜ , 2

∇ φ = 0, ∇φ · ∇h = 0,

on ΓFS

(1a)

in

(1b)

Ω

on Γb

(1c)

where h(x) : Γb → R describes variation in the still water depth, however, is assumed constant in this work. The temporal evolution of the water surface is described by z = η(x,t) : ΓFS × T → R. The notations are illustrated in Figure 1. The unsteady free surface boundary conditions can in the MEL formulation be described as Dx = ∇φ , in Ω Dt Dη = w, on ΓFS Dt 1 Dφ = ∇φ · ∇φ − gz − p, Dt 2

(2a) (2b) on ΓFS

(2c)

where ’∼’ is used to denote free surface variables defined at z = η, and g is the gravitational acceleration defined to be g = 9.81 m/s2 . We express the free surface equations in terms of free surface variables only (see derivation in [17]) in one space dimen2

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sion of the free surface ΓFS in the form

Bernoulli’s equation

Dx ∂ φ˜ ∂η = − w˜ , (3a) Dt ∂x ∂x Dη = w, ˜ (3b) Dt !  2   1 ∂ η ∂ φ˜ ∂ φ˜ ∂η 2 Dφ˜ = − 2w˜ + w˜ 2 + w˜ 2 − gη, Dt 2 ∂x ∂x ∂x ∂x

p 1 = −gz − φt − ∇φ · ∇φ , ρ 2

where we have introduced the convenient notation φt = ∂ φ /∂t for the Eulerian rate of change of φ . In the MEL setting as seen from the reference frame of a moving body, we obtain the pressure equation

(3c)

1 p = −gz − φt − ∇φ · ∇φ + vo · ∇φ , ρ 2

having assumed a zero reference pressure at the free surface, cf. eq. (2c). To reach this formulation we used that spatial and temporal differentiation of free surface variables are given by the chain rules for differentiation ∂ φ ∂ φ ∂η ∂ φ˜ = + , ∂x ∂ x z=η ∂ z z=η ∂ x ∂ φ˜ ∂ φ ∂ φ ∂η = + . ∂t ∂t z=η ∂ z z=η ∂t

on

Γ\(ΓFS ∪ Γbody )

(4a)

Dφ ∂φ ≡ + vo · ∇φ . Dt ∂t

on Γbody

for a body reference frame co-moving with the body at speed v0 . This is useful when φt is approximated by backward finite difference stencils, and have been used to verify the implemented BVP solver for the acceleration potential described next. Wave-body motion using an acceleration potential method To consider the wave-body interaction problem that describes the coupled interaction between fluid motion and free body motion, we need to have a way of determining the φt term in the pressure eq. (9). A general way, to find an approximation to this quantify is to use a method that follows the original idea of Vinje and Brevig (1981) [21] and later generalized by Tanizawa (1995) [18] of introducing a nonlinear acceleration potential. Methods based on introducing an acceleration potential are known to be an accurate and general approach to solve the problem of coupling body motions with fluid motions. The approach is particular useful for freely moving bodies subject to finite amplitude motions. In the following, we explain the method for translational movements of a body. By applying the gradient operator to eq. (9) we derive the Euler’s equation describing ideal fluid motion

(5)

(6)

where vo is the body velocity vector with respect to the gravity center of the body, and n is the body surface normal vector. For fixed bodies (not moving) vo = 0. For moving (non-stationary) bodies this body boundary conditions needs to be updated during a simulation. Either this is done by prescribing analytically the velocity leading to forced motion, or by updating the net force on the body boundary Γbody . At this boundary, the fluid particles are constrained to move along the tangential direction and to follow the body motion according to n·

Dx = n · vo . Dt

(10)

(4b)

where we have introduced the fluid-body boundary (Γbody ). On any body, fixed, fully or partially submerged inside the fluid, the body boundary condition is written as φn = n · vo ,

(9)

having used that the connection between the material (comoving) derivative and the Eulerian derivative is defined as

Boundary conditions The boundary conditions are assumed to be rigid boundaries everywhere but the free surface of the fluid domain, i.e. stated in the form of a no-flux condition as φn = n · ∇φ = 0,

(8)

D∇φ 1 1 = − ∇(∇φ · ∇φ ) − ∇p − g∇z. Dt 2 ρ

(11)

Since, the velocity field vector is defined as u ≡ ∇φ , the acceleration of the fluid is related to the velocity as

(7a)

The pressure on the instantaneous wetted surface of the body that takes into account the nonlinear dynamic terms, is given by

a≡ 3

Du ∂ ∇φ = + (∇φ · ∇)∇φ , Dt ∂t

(12)

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where we have used the definition of eq. (10). Considering the relation from vector calculus 1 ∇(u · u) = (u · ∇)u + u × (∇ × u), 2

The kinematic boundary condition [18] for the acceleration of a body that only undergoes translational movement (i.e. no angular accelerations stemming from rotations) is (13)

Φn = n · (a0 ) − kn (∇φ − v0 )2 ,

together with the potential flow assumption of irrotational flow ∇ × u = 0, the fluid acceleration eq. (12) can be expressed in the form  a=∇

 ∂φ 1 + (∇φ )2 . ∂t 2

∂φ 1 + (∇φ )2 , ∂t 2

(14)

mao = m˙vo + ω × mvo = Fg + F p ,

(16)

ˆ Fg = −mgk,

Using the irrotational assumption in the form ∇ × ∇φ = 0 it is possible to conclude that ∇ × Φ = 0 implying that the acceleration potential is also irrotational. However, we also have ∇2 Φ = ∇2



1 (∇φ )2 2

(17)

Φn = n · ∇Φ = n · a,

on Γ\ΓFS

(18a)

on Γ\ΓFS

(18b)

on

ΓFS

Γbody

ˆ pnds = Fpx iˆ + Fpz k,

(22)

(23a) (23b) (23c) on Γbody (23d)

At every instant of time, this boundary value problem is solved for φt after the Laplace problem for φ has been solved. This makes it possible to update the boundary conditions that depends on φ before computing φt .

where in two space dimensions n = nx iˆ + nz kˆ is the unit normal vector of the body surface, V and a are the velocity and acceleration vectors of the surface, respectively. At stationary walls and the sea bed we have V = 0 and at a moving body surface V = vo . The free surface boundary condition for the acceleration potential Φ is determined from eq. (8), which at the free surface states (p = 0 assumed) Φ = −gη,

Z

1 φt = −gη − (∇φ )2 , on ΓFS 2 ∇2 φt = 0, in Ω 1 n · ∇φt = − n · ∇(∇φ )2 , on Γ\ΓFS 2 1 n · ∇φt = n · a0 − kn (∇φ − vo )2 − n · ∇ (∇φ )2 , 2

showing that the acceleration potential does not satisfy the Laplace equation due to the nonlinear term. Thus, we need to find a suitable way to use the acceleration potential in order to have a means to determine the φt that will be needed for determining pressure (cf. eq. (8)) and forces acting on bodies. The kinematic boundary conditions for velocity and acceleration potentials are φn = n · ∇φ = n · V,

Fp =

where iˆ and kˆ are the unit vectors for the Cartesian x- and zdirections. We assume ω = 0 to only deal with translational movement of the body. To summarize, we exploit the acceleration potential function Φ to formulate a Laplace problem for φt (x, z,t) : Ω × T → R satisfying

 6= 0,

(21)

where m is the body mass, ao is the acceleration of the center of gravity, ω is the angular velocity, and the right hand side is the net force vector acting on the body expressed in terms of the gravitational and wave-induced induced forces

(15)

and we can express the acceleration as a ≡ ∇Φ.

(20)

where a0 is the translational acceleration and v0 is the translational velocity. kn = −1/R is the normal curvature of the body surface with R the radius of the cylinder. The equation for the center of gravity of the body is given by

Now, we can introduce the nonlinear acceleration potential Φ≡

on Γbody

NUMERICAL DISCRETIZATION In the following, we present the discretization of the governing equations in the general framework of spectral element methods expressed first in terms of weak formulations and then generate the discrete equations that constitute the numerical free surface wave model.

(19) 4

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We start by forming a partition of the domain to obtain a tessellation ThFS of the free surface (FS) boundary ΓFS consisting of Nel non-overlapping shape-regular elements ThFS such that the union of all of these elements is a complete representation of Nel ThFS,k = ThFS with k denoting the the free surface area, i.e. ∪k=1 k’th element. We introduce for any tesselation Th the spectral element approximation space of continuous, piece-wise polynomial functions of degree at most P, V = {vh ∈ C0 (Th ); ∀k ∈ {1, ..., Nel }, vh |Thk ∈ PP },

where the following global matrices have been introduced Mi j ≡

Z

Mibj ≡

Z

(Axb )i j ≡

Z

ThFS ThFS ThFS

Ni N j dx,

(28a)

b(x)Ni N j dx,

(28b)

b(x)Ni

dN j dx, dx

(28c)

(24) FS

where fh ∈ RN is a vector containing the set of discrete nodal values on the mesh used for the discretization of the free surface boundary. The starting point for the spatial discretization of the Laplace equation is the weak formulation that is expressed as : find φ ∈ V such that

which is used to form finite-dimensional nodal spectral element approximations N FS

fh (x,t) =

∑ fi (t)Ni (x),

(25)

i=1 FS

where {Ni (x)}Ni=1 ∈ V is the global finite element basis functions with cardinal property Ni (x j ) = δi j at mesh nodes with δi j the Kronecker Symbol.

Z Th



ThFS

L φh = b,

∂η ∂ φ˜ − w˜ vdx, ∂x ∂x

Li j = −

∇φ · ∇vdz = 0,

(29)

L ∈ Rn×n ,

b ∈ Rn

(30)

2D Z Nel

Z Th

(∇N j ) · (∇Ni )dx = − ∑

k k=1 Th

(∇N j ) · (∇Ni )dx.

Likewise, for the Laplace problem based on the acceleration potential, we can reuse the discretization of the Laplace operator, but the boundary conditions are different, since both different free surface, sea bed, and body-boundary conditions are different. The right hand side vectors b and c are updated to take into account inhomogeneous Dirichlet and Neumann boundary conditions. The numerical discretization leads to

Dxh = Ax φ˜x − Axw˜ h ηh , (27a) Dt Dηh = w˜ h , (27b) Dt   Dφ˜h 1 w˜ 2 (ηx ) (φ˜ ) = −M g ηh + Ax x h φ˜h + M w˜ h w˜ h + Ax h h ηh M Dt 2 M

φ˜h ,

Th

(31)

for all v ∈ V . If we substitute the expressions in (25) into eq. (26) and choose v ∈ V , the discretization in one spatial dimension becomes

w˜ h (ηx )h

Z

where

(26c)

− Ax

∂ Th

vn · ∇φ dx −



Dx vdx = (26a) Dt ThFS Z Z Dη vdx = [w]vdx, ˜ (26b) ThFS Dt ThFS !# "  2 Z Z ∂ φ˜ ∂ η ∂ φ˜ Dφ˜ 1 vdx = − 2w˜ vdx ∂x ∂x ∂x ThFS Dt ThFS 2 ! # " 2  Z 1 ∂ η + w˜ 2 + w˜ 2 − gη vdx, ∂x ThFS 2 Z

I

for all v ∈ V where the boundary integrals vanish at the domain boundaries. The numerical discretization leads to the linear system

Weak formulation of the free surface conditions The weak formulation of the free surface conditions is derived in [17] and reproduced here. Find f ∈ V where f ∈ x, η, φ˜ such that Z

∇ · ∇φ vdx =

L (φt )h = c,

L ∈ Rn×n ,

c ∈ Rn

(32)

where the solution (φt )h makes it possible to compute the hydrodynamic wave-induced forcing everywhere in the fluid domain. Specifically, it is needed to update the forcing on the body, to account for the fluid-body interactions.

(27c) 5

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FIGURE 2: Illustration of a hybrid mesh for representation of the fluid domain. The mesh consists of a curvilinear quadrilateral free surface layer combined with an unstructured curvilinear triangulation of a floating cylindrical body with a curved bottom surface. This example highlight the flexibility in the use of geometrically adapted unstructured meshes and is using a polymomial order P = 4 mesh giving the degrees of freedom associated with the quadrature points.

Wave absorption To absorb the radiating waves resulting from the body movements we use the embedded penalty method described in [4]. The resulting free surface boundary conditions used are Dηh 1 − γ(x) = Nη (ηh , φ˜h ) − ηh , Dt τ 1 − γ(x) ˜ Dφ˜h = Nφ˜ (ηh , φ˜h ) − φh , Dt τ

R and a circular segment at the bottom with radio R whose center C is located above of the top lid in the vertical line that divides the solid by the middle. Based on the shallow water assumption Lannes presented an analytical formula for the nonlinear problem of the water-body contact points x± [19]:

(33a)

  2 R η(t, x± ) = τ0 − h, √ w(t) 2 g   h 1 √ h +C(r) + τ0 (r) = , 3 C(r) 1/3 p 3 C(r) = −4r + 2r0 + 4 r(r − r0) , 2

(33b)

where Nη and Nφ˜ are the right hand side functions and the sponge layer (relaxation) functions γ ∈ [0, 1] for the sponge layer zone, can be chosen to have minimal numerical artifacts at sponge layer zone interfaces according to the analysis given in [22]. The relaxation parameter is defined to be equal to the time steps size, i.e. τ = ∆t.

(34a) (34b) (34c)

where h is the still water depth, w(t) is the body vertical velocity 4 1/3 such that v0 = (0, w(t)) and r0 = 27 h . In the following we use R = 10m and initial position of the center ZC = 4.57m. The body is subjected to a harmonic oscillation of period T = 10s and amplitude B = 2m in a water depth h = 15m. The SEM discretization consists of 356 high-order (P = 6) elements with a top quadrilateral layer above an unstructured triangular part. The triangular elements are refined close to body. The mesh is updated in order to follow the moving body and Laplace smoothing is applied to prevent the elements from becoming distorted. The time step used is T /60 ≈ 0.03s and a very mild modal filter (removing 5% of the top mode of a hierarchical Legendre polynomial basis) is applied at every RK sub-step. In Figure 3, the free surface elevations corresponding to the SEM solutions at four time instances are shown. The shown surface elevations visually agree favourable to [19, Fig. 7]. Remark, that [19] uses a numerical model based on the shallow water equations and thus an exact match is not to be expected. Figure 4 shows the time history of the computed water-body contact points x± compared to the analytic solution. Overall the solution is matching while the main differences are (i) the slight

NUMERICAL EXPERIMENTS In [16, 17] it was demonstrated how a quadrilateral layer (in 2D) below the free surface elevation help stabilize the spectral element model for both wave propagation and wave-structure interactions with fixed (non-moving) structures. In this work, we use again such a quadrilateral layer below the free surface, and form a hybrid mesh using an unstructured triangulated mesh to resolve an arbitrarily shaped body below the free surface quadrilateral layer. Using high-order elements the body surface is represented using curvilinear elements, as is the free surface elevation. See Figure 2. Surface-piercing body in forced vertical motion A surface-piercing body in prescribed vertical motion is considered following the set-up in Lannes [19]. The body consists of the union of a rectangular box of width 2R and height 2R sin (π/3) − 6

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lagging of the numerical solution and (ii) the deeper through. Again, we recall that the analytical solution is based on the shallow water equations which only retain the nonlinearity within the hydrostatic assumption. The FNPF model retain all nonlinear terms including the non-hydrostatic components.

1.5

1

0.5

0 time =0.08

time =3.33

10

10

5

5

0

0

-5

-5

-10

-10

-0.5

-1

-1.5 0

-15 -100

-80

-60

-40

-20

0

20

40

60

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100

-15 -100

-80

-60

-40

-20

(a)

0

20

40

60

80

0.5

1

1.5

2

2.5

3

100

(b)

FIGURE 4: Forced motion of floating cylinder. Time history of time =6.67 10

5

5

0

0

-5

-5

-10

-10

-15 -100

-80

-60

-40

-20

0

(c)

20

water elevation at water-body contact points. A comparison is made between the analytical solution of Lannes [19] obtained in the setting of shallow water equations and the FNPF results obtained using the SEM model for same setup in a numerical wave tank.

time =10.00

10

40

60

80

100

-15 -100

-80

-60

-40

-20

0

20

40

60

80

100

(d)

the computational costs associated with mesh updates are negligible compared with the solution of Laplace problems, we rely on high-order elements for accuracy in the numerical solution.

FIGURE 3: Forced motion of floating cylinder. Free surface pro-

file at four different instants in the first period of the motion.

We are interested in estimation of the wave-induced vertical loads that result from the forced motions. Following [23] we decompose the non-dimensional vertical force in a Fourier series in the form

Submerged cylinder in forced vertical motion We consider the forced heave motion of a cylinder fully submerged below the free surface undergoing large amplitude motion. For the test case here we follow the set-up described in [20] with a cylinder of radio R = 0.1m and initial submergence zc = −3R in prescribed vertical motion of fixed dimensionless wave number kR = 0.1 and variable amplitude B in a water depth of h = 3m. The computational mesh is again of hybrid type and consists of 180-200 elements and use local polynomial series expansion orders of P = 6. The body boundary condition is prescribed using eq. (18a) accounting for the forced motion of the body. The movement of the body makes it necessary to update the mesh between time steps, and this is done by displacing waterbody nodes with the body and employing a Laplace smoothing technique to avoid excessive distortion of triangular elements. When the body is closer to the free surface we switch to a mesh configuration where the cylinder is embedded in the quadrilateral top layer. This mesh strategy allows for a good triangle quality measure at all times and it is efficient in the sense that

Fz (0) (n) = Fz + ∑ Fz sin(nω + ψ (n) ). ρBπR2 ω 2 n≥1

(35)

In figure 5, we compare the numerical results obtained using the SEM solver against the numerical results of [23] and [20] showing excellent agreement for the cases of B/R = 0.2, 0.4, 0.6, 0.8, 1, 1.25. Larger values of B/R lead to meshrelated instabilities in the current model setup and needs further improvement in the ongoing work. Dimensionless fluid velocity profiles are shown in figure 6 along with mesh vertices in the vicinity of the cylinder for the case B/R = 1 at constant time intervals of T /6 in a complete period of the motion. The model retains symmetry about x = 0 throughout the simulation as expected. Please note the switch between the above mentioned mesh configurations. 7

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Guerber Wu SEM

Guerber Wu SEM

0.3

1.2 0.2

0.05

0.1

Guerber Wu SEM

0

0.5

1

1.5

0

0.5

(a) Mean value

1

1.5

0.5

(b) 1st Harmonic amplitude

1

1.5

(c) 2nd Harmonic amplitude

FIGURE 5: Forced motion of submerged cylinder. Harmonic coefficients of the non-dimensional vertical force with kR = 0.1. 0.2

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-2R

-R

0

x/h

x/h

R

2R

3R

4R

x/h

/h

(b) Vertical velocity 0.2

0.2

0.2

0.2

0.2

0.2

0.2

0

0

0

0

0

0

0

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.6 -4R

-3R

-2R

-R

0

R

x/h

2R

3R

4R

-0.6 -4R

-3R

-2R

-R

0

R

2R

x/h

3R

4R

-0.6 -4R

-3R

-2R

-R

0

R

2R

3R

4R

-0.6 -4R

-3R

-2R

-R

x/h

0

R

2R

3R

4R

-0.6 -4R

-3R

-2R

-R

0

x/h

x/h

R

2R

3R

4R

-0.6 -4R

-3R

-2R

-R

0

x/h

R

2R

3R

4R

-0.6 -4R

-3R

-2R

-R

0

R

2R

3R

4R

x/h

(c) Mesh detail near cylinder

FIGURE 6: Forced motion of submerged cylinder. Dimensionless velocity profiles and corresponding hybrid mesh vertices in one period

for the case kR = 0.1 and B/R = 1. Conclusion

duction of the number of elements needed, e.g. in comparison with the number of elements needed in classical FEM methods for FNPF equations [24, 25] to resolve features in depth and curvature.

A new stabilised spectral element FNPF numerial model is extended to include moving bodies. Through numerical experiments it is demonstrated how a cylindrical-shaped body undergoes prescribed forced motion and results are compared against results from other models and shows excellent agreement. The spectral element model uses high-order elements that helps reduce the total number of elements and as a consequence the total degrees of freedom in the discretization through significant re-

In ongoing work, the focus is on improving the efficiency of the linear solver, the mesh update procedures and extend the work to three space dimensions through extension of [26] to also include moving bodies. These ongoing developments will make it possible to target engineering applications. 8

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ACKNOWLEDGMENT This work contributes to the activities in the research project Multi-fidelity Decision making tools for Wave Energy Systems (MIDWEST) that is supported by the OCEAN-ERANET program. The DTU Computing Center (DCC) has supported the work with access to computing resources.

[12] Patera, A. T., 1984. “A spectral element method for fluid dynamics: Laminar flow in a channel expansion”. J. Comput. Phys., 54, pp. 468–488. [13] Xu, H., Cantwell, C. D., Monteserin, C., Eskilsson, C., Engsig-Karup, A., and Sherwin, S., 2018. “Spectral/hp element method: recent development, applications, and perspectives”. J. Hydrod., 30(1), pp. 1–22. [14] Robertson, I., and Sherwin, S., 1999. “Free-surface flow simulation using hp/spectral elements”. J. Comput. Phys., 155(1), pp. 26–53. [15] Westhuis, J.-H., 2001. “The numerical simulation of nonlinear waves in a hydrodynamic model test basin”. PhD thesis, Department of Mathematics, University of Twente, The Netherlands. [16] Engsig-Karup, A.P., Eskilsson, C., and Bigoni, D., 2016. “A stabilised nodal spectral element method for fully nonlinear water waves”. J. Comput. Phys., 318(6), pp. 1–21. [17] Engsig-Karup, A.P., Monteserin, C., and Eskilsson, C. “A stabilised mixed Eulerian Lagrangian spectral element method for nonlinear wave interaction with fixed structures”. Submitted for peer-review. See also URL http://https://arxiv.org/abs/1703.09697. [18] Tanizawa, K., 1995. “A nonlinear simulation method of 3-D body motions in waves”. Jap. Soc. of Naval Arch., 178(1), pp. 179–191. [19] Lannes, D., 2017. “On the dynamics of floating structures”. Ann. PDE, Springer, 3(1), pp. 1–81. [20] Guerber, E., Benoit, M., Grilli, S.T., and Buvat, C., 2012. “A fully nonlinear model for wave interactions with submerged structures in forced or free motion”. Eng. Anal. Bound. Elem., 36(7), pp. 1151–1163. [21] Vinje, T., and Brevig, P., 1981. “Numerical simulation of breaking wave”. Adv. Water Resour., 4(2), pp. 77–82. [22] Engsig-Karup, A. P., 2006. “Unstructured nodal DG-FEM solution of high-order boussinesq-type equations”. PhD thesis, PhD. Thesis. Department of Mechanical Engineering, Technical University of Denmark. [23] Wu, G., 1993. “Hydrodynamic forces on a submerged circular cylinder undergoing large-amplitude motion”. J. Fluid Mech., 254, pp. 41–58. [24] Wu, G., and Taylor, R. E., 1994. “Finite element analysis of two-dimensional non-linear transient water waves”. Appl. Ocean. Res., 16(6), pp. 363 – 372. [25] Wu, G., and Taylor, R. E., 1995. “Time stepping solutions of the two-dimensional nonlinear wave radiation problem”. Ocean Engng., 22(8), pp. 785 – 798. [26] Engsig-Karup, A. P., Eskilsson, C., and Bigoni, D., 2016. “Unstructured spectral element model for dispersive and nonlinear wave propagation”. In Proceedings of 26th ISOPE 2016, Greece.

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