Oblique Wave-Induced Responses of A VLFS Edged with A Pair of Inclined Perforated Plates

This paper is concerned with the hydroelastic responses of a mat-like, rectangular very large floating structure (VLFS) edged with a pair of horizontal/inclined perforated anti-motion plates in the context of the direct coupling method. The updated Lagrangian formulae are applied to establish the equilibrium equations of the VLFS and the total potential formula is employed for fluids in the numerical model including the viscous effect of the perforated plates through the Darcy’s law. The hybrid finite element-boundary element (FE-BE) method is implemented to determine the response reduction of VLFS with attached perforated plates under various oblique incident waves. Also, the numerical solutions are validated against a series of experimental tests. The effectiveness of the attached perforated plates in reducing the deflections of the VLFS can be significantly improved by selecting the proper design parameters such as the porous parameter, submergence depth, plate width and inclination angle for the given sea conditions.


Introduction
The concept of very large floating structure (VLFS) is introduced as an alternative option of pursuing an environmentally friendly and sustainable technology in birthing land from the ocean and recognizing the rising water level due to the global warming. For a typical pontoon-type VLFS, a noticeable feature is the very great ratio of the horizontal scale relative to the vertical scale so that the interaction between the elastic deformation and the fluid flow field around it is necessarily considered in order to obtain the vertical responses of a VLFS. For most applications, the satisfaction is directly related to the hydroelastic response of a VLFS. Thus, various ways (Wang et al., 2010) have been proposed to reduce the wave forces and hydroelastic responses of VLFS.
An interesting approach of mitigating hydroelastic responses of a VLFS involves attaching auxiliary beam at the fore-end of the VLFS as a sacrificial floating structure to dissipate the incident wave energy. Kim et al. (2005) de-veloped the idea of reducing the motion of the main parts of the articulated floating structure by allowing the other parts of the structure to move more. Fu et al. (2007), Gao et al. (2011) and Zhao et al. (2015) further analyzed the deflection, bending moments and shear forces at unit connections and examined both the effects of the connector location and stiffness in reducing the hydroelastic responses of the VLFS. Another attractive option for reducing the effect of surface waves on the VLFS is to attach a submerged antimotion device to the fore-end of the VLFS. These anti-motion devices are classified into the submerged vertical (Ohta et al., 1999;Masanobu et al., 2003;Takagi et al., 2000) and horizontal non-perforated plates (Ohta et al., 1999;Watanabe et al., 2003;Pham et al., 2008;Cheng et al., 2017), which can dissipate the incident wave energy, i.e. transmission from the outside to the inside of the VLFS. Actually, the surface wave energy by attaching perforated plates can be further reduced owing to viscous dissipation of the porosity. In the past years, analytical solutions for wave interaction with perforated plates have been given by some researchers such as Wu et al. (1998), Cho and Kim (2008), and Cho (2016). They applied Darcy's law of linear relation between the velocity and pressure drop to represent energy dissipation. The quadratic relation between traversing velocity and pressure differential in the case of perorated or slotted structures was also considered by Molin (2001Molin ( , 2011. Liu et al. (2012) and Cho et al. (2013) further employed an alternative method without finding complex roots to wave interaction with the single or dual perforated plates. These analytical solutions are compared with the numerical results by using the developed multi-domain boundary element method (BEM) with simple Green functions. Following this idea, Cheng et al. (2014Cheng et al. ( , 2015Cheng et al. ( , 2016aCheng et al. ( , 2016b studied analytically and numerically the effectiveness of horizontal perforated plates in reducing hydroelastic responses of the VLFS. They also conducted systematic model experiments with various plate porosities and developed the empirical relationship between plate porosity and the porous parameter by a least-squares fitting technique. Until now, most of the investigations on the hydroelastic analysis of a VLFS have adopted the modal expansion method as the primary method, which mainly consists of four steps i.e. calculation of the dynamic modes of a structure, solution of the diffraction and radiation velocity potentials, evaluation of the corresponding generalized coefficients, and construction of the generalized equations of motion. However, the resulting analysis capability is limited to the solution for the linear hydroelastic problems of floating structures, i.e. method of eigenfunction expansions (MEE) (Watanabe et al., 2003;Pham et al., 2008, Cheng et al., 2016b). An alternative direct coupling method (Eatock Taylor, 2007;Kim et al., 2013Kim et al., , 2014Yoon et al., 2014) has been proposed recently, in which the structural and water equations are directly coupled with each other, and the coupled plate-water equations are calculated simultaneously. In contrast to the modal expansion method, this method does not require the effort of calculating the added mass, radiated damping and wave excitation forces one by one. The interaction term is more easily constructed, and its components are more sparsely stored. Thus, the computational algorithm is more simple and straightforward.  developed a tight coupling formula for three-dimensional hydrodynamic analysis of floating flexible structures subjected to surface regular waves as well as other excitation loads. In their solution, the direct coupling method can be directly extended to the nonlinear analysis of floating structures. Kim et al. (2014) further extended the theory to construct the hydroelastic response contours and proposed a preliminary design procedure for the VLFS using the hydroelastic design contours in irregular waves. Most recently, Yoon et al. (2014) have obtained a numerical solution for floating elastic plates with multiple hinge connections by directly coupling the total potential-based bound-ary element method (BEM) and the structural deformationbased finite element method (FEM).
In this paper, we investigate the hydroelastic responses of a VLFS edged with a pair of inclined perforated anti-motion plates following the direct coupling method proposed by Eatock Taylor (2007). In this analysis, we establish the numerical model and scheme similar to Kim et al. (2014) and Yoon et al. (2014). However, the formula is extended to include the anti-motion plates under oblique incident wave conditions. In addition, the non-conforming quadraticserendipity (NC-QS) Mindlin plate elements (Tay et al., 2007) instead of mix interpolation of tensorial components (MITC) plates are employed to discretize the VLFS and anti-motion plates. Such a Mindlin plate element can produce good results with less computational effort. It does not suffer from spurious modes and shear locking phenomenon, and considers the effects of shear deformation and rotary inertia, which are neglected in the classical thin plate theory. As for fluid part, the total velocity potential formula is applied for surrounding fluid in context of three-dimensional potential theory. A hybrid finite element-boundary element (FE-BE) method is applied to obtain numerical solution, where the FEM is used to discretize the linearized structural equation whereas the BEM is used to handle the fluid integral boundary equation. A series of experiments with a sandwich-type VLFS model with attached inclined perforated anti-motion plates are also conducted in the wave basin to confirm the numerical model and results. After checking the reliable correlation between the predicted and measured results, the dependence of the hydroelastic responses on various wave and plate parameters is systematically investigated through an extensive parametric analysis. Finally, the main conclusions of this paper are drawn.

Plate-water model
The sketch of hydroelastic behavior in oblique incident waves of a box-shaped VLFS edged with a pair of inclined perforated submerged plates is given in Fig. 1. A Cartesian coordinate system with the z-axis upward, and the x-y plane at the mean position of the free surface, is adopted for the mathematical description, and the water depth is denoted by h. The VLFS has a length L, width B, height h s , and d is the draft of the VLFS in the z direction. The perforated anti-motion plates of the width 2a and length B are attached at both ends of the VLFS at the submergence depth d 1 . The plate volume is denoted by V, and the whole fluid domain is defined at Ω which contains the wet surface of the VLFS S b , the surface of the submerged inclined plates S a1 and S a2 , the free surface S f , the seabed S d and the infinite cylindrical surface S ∞ . The heading angle of monochromatic incident waves with respect to the x-axis is θ. It is assumed that the plate has homogeneous, isotropic, and linear elastic materi-al, and the fluid is irrotational, and ideal so that linear potential theory can be used.

Equations of motion for VLFS
The updated Lagrangian formula ) is used to obtain the formulae corresponding to both the hydrostatic and the hydrodynamic analysis. Thus the equilibrium equations of the VLFS at time t can be expressed as: where σ ij denotes the Cauchy stress tensor at time t; ρ s is the structural density at time t; g is the acceleration of gravity; U i is the vertical displacement at time t; δ ij is the Kronecker delta; n i is a unit normal vector (the positive direction points out of the fluid domain); the subscripts i and j vary from x to z to express the components of tensor, and the Einstein summation convention is adopted. The fluid pressure P comprises the hydrostatic and hydrodynamic pressure at the bottom of the VLFS, i.e.
where ρ w is the fluid density; P d is the hydrodynamic pressure. The principle of virtual work for the VLFS at time t is where δU i and δE ij represent the virtual displacement vector and linear strain tensor, respectively. The boundary surface S B includes the wet surface of the VLFS S b , and the surface of the submerged inclined plates S a1 and S a2 . Eq. (4) represents the structural motion formula at time t, and it includes two steps, that is the hydrostatic analysis focused on the static equilibrium position and the hydrodynamic solution treating the dynamic equilibrium position. And then we linearize Eq. (4) at the static equilibrium state and subtract the hydrostatic equilibrium denoted by time t=0, we can obtain the steady state equation (Kim et al., 2014;Yoon et al., 2014) − where C ijvl is the stress-strain relation tensor (v and l also vary from x to z), subscript 0 denotes time t=0 and , , in which the superscript i 2 =-1, Re represents the real part of the argument, and ω refers to the angular frequency of wave.

Equations of motion for fluid
With the steady state motions of the harmonically excited system at a circular frequency ω, the velocity potential of the water can be expressed as . The spatial velocity potential is governed by the Laplace's equation and the boundary conditions where r denotes the radial coordinate measured from the center of the VLFS; k is the wave number at the constant depth h; represents the velocity potential at the positiond 1 ±; represents the incident potential: where k x =kcosθ and k y =ksinθ are the x-component and ycomponent wavenumber, respectively. A is the incident wave amplitude. The imaginary part of the complex-valued frequency dependent parameter τ in Darcy's model shown in Eq. (11) represents the inertia effect and it can be neglected when the submerged plate is thin and the hole sizes are small. However, the positive real part of τ is called porous-effect parameter, which is related to the porosity coefficient and viscous dissipation and can be obtained from experiment. According to Cho and Kim (2008), the relationship between porous parameters b and τ can be expressed as: When b=0, the plate corresponds to an impermeable plate; while for b=∞, it means that the plate is transparent. After systematic model tests on the effectiveness of the perforated plates with various porosities, Cheng et al. (2015) developed the following empirical relationship between the actual porosity P and porous parameter b in the range of the porosity P=0.02-0.4: (14) This empirical formula can be applied to the extensive parametric investigations on the hydroelastic responses of the VLFS edged with perforated plates. With the help of Green's second identity, the prescribed boundary value problem i.e. Eqs. (6)-(11) can be transformed into the following boundary integral equation: where G is the free-surface Green's function for water of finite depth and satisfies the seabed boundary condition, water free surface boundary condition and boundary condition at infinity; the subscript ξ indicates that the integral should be conducted with respect to ξ. The velocity potential is related to the fluid pressure p on the bottom surface of the VLFS from the linearized Bernoulli's equation Substituting Eqs. (10), (11) and (16) into Eq. (15), we have (17) By premultiplying both sides of Eq. (17) by δp and integrating over the wet surface S B0 , S a10 and S a20 , the as-sembly result takes the form

Discretization of the coupled equations
Term-by-term finite and boundary element discretizations of the linearized structural Eq. (5) and the fluid Eq. (18), respectively, the coupled matrix form for the steady state 3D hydroelastic analysis can be obtained as: where "ˆ" denotes the vector, i.e. and are the unknown displacement and pressure vectors, respectively, and is the vector related to incident potential. These sub-matrices and sub-vectors are defined as follows: where S M , S K and C up are the mass matrix, stiffness matrix and fluid-structure interaction matrix, respectively. F M and F G are the influence matrixes related to the boundary element discretization of fluid Eq. (18).
In order to use the finite element method (FEM) to solve the plate equation, the floating plate is approximated by a number of non-conforming quadratic-serendipity (NC-QS) Mindlin plate elements. This study applies the reduced integration method and the superposition of non-conforming modes to prevent shear locking and spurious modes phenomena. Detailed procedures are described by Tay et al. (2007). The water equation is solved for the fluid pressure by using the boundary method (BEM). Thus the body surfaces of VLFS and anti-motion plates are discretized with a number of constant elements, where physical variables are interpolated by the shape functions (Teng and Eatock Taylor, 1995;Ji et al., 2017).

Modal analysis
To reduce the number of unknowns, we adopt the mode superposition method, which expands the response of the VLFS by a series of the products of the eigenvector ψ and the generalized coordinate vector ζ: where I and are given by and . Then the modified plate-water coupled equation is Upon solving the coupled plate-water Eq. (22), we obtain the generalized coordinates ζ m and then we back-substitute these generalized coordinates into Eq. (20) to solve the plate deflection.

Experiments
In order to confirm the numerical solutions based on the theory described so far, we carried out a series of experiments in a three-dimensional basin (40 m long, 8 m wide, and 1.1 m deep) located at Dalian University of Technology. The square basin is equipped with a piston type wave maker capable of producing regular and irregular waves. The wave period range used in our experiment is from 0.599 s to 3.416 s and the wave amplitude is maintained around 20 mm. The incident wave is generated by a user-defined timevoltage input to the wave maker to the VLFS model at incident directional angle θ=0°. The water depth is fixed at 0.6 m. The sandwich-type VLFS model is placed at 25 m from the wave maker, and is made of aluminum plate and polyethylene foam, as shown in Fig. 2. The main material of the model should be aluminum plates (8 m in length, 1.2 m in width and 1.8 mm in thickness) with a sufficient weight and rigidity. The middle polyethylene foam (8 m in length, 1.2 m iin width and 10 mm in thickness) between the upper and lower aluminum plates can adjust the rigidity by changing the height of the polyethylene foam without changing the weight. The polyethylene foam (8 m in length, 1.2 m in width and 60 mm in thickness), which has negligible weight and rigidity at the bottom of the VLFS model, is used as a buoyancy material and can transfer the fluid forces to the aluminum plates. In this way, since the upper plate of the VLFS model is located at a sufficient distance from the water surface, such difficulties as deck wetness that may be encountered in using a simple thin plate model can be avoided. Table 1 summarizes the principal details of the VLFS model and wave characteristics used in our experiment.
The anti-motion device with a pair of perforated plates is installed at the desired submergence depth to four vertical steel frames clamped to both ends of the VLFS model. The circular holes are distributed uniformly to represent the given five different porosities (P=0, 0.038, 0.072, 0.11, 0.18). The perforated plate is installed at submergence depths of d 1 =10, 15 and 20 cm, respectively. The length, width and thickness of the perforated plate are 1.2 m, 32 cm and 5 mm, respectively. The inclination angles β=4° and 9°o f the perforated plates are chosen throughout our experiments.
In order to prevent drifting in waves, two models of mooring devices were made from standing steel frames in the water basin as shown in Fig. 3: the one is the chain model (four 200-cm-long wire rope-springs with universal joints at both ends are attached to the middle location of the VLFS and the standing steel frames, which represent the mooring dolphins), the other is the fender model (ten 25-cm-long wire rope-springs are arranged at the side of the VLFS model).
The vertical displacements can be measured accurately at various points with the displacement sensor having an accuracy of 0.1 mm, in which the instantaneous positions of the measured points of the upper deck are detected by a data acquisition system. Twenty-one sites are selected for elastic response measurements, as indicated in Fig. 4, where the data for the fore-end, mid-position, and the back-end of the VLFS are from the averaged data (#1+#12+#17)/3, (#6+#14+ #19)/3 and (#11+#16+#21)/3, respectively.

Numerical results and discussions
Since the mode superposition method was used in the present computations, we first carried out a mode test on the VLFS attached with a pair of perforated plates for the case: kL/2=31.416, 2a/L=0.04, 2d 1 /L=0.025, b 1 =5.562, θ=60° and the major details of the VLFS are described in Section 4. Fig. 5 shows the longitudinal distribution of deflections along the centerline of the structure. In this analysis, the element size to wavelength ratio must be smaller than 0.1 after checking the convergence, and the vertical deflection obtained is confirmed to be stable and periodic after three cycles of the wave period. We found that modes M=27 are sufficient for the deflections to converge of the VLFS (Here, the first 3 modes represent the rigid motions and the 4th is the first elastic mode) and thus are adopted in the present study.
Next, the numerical solutions are compared against the experimental results for various relevant porosities. In this analysis, the element size to wavelength ratio must be smaller than 0.1 after checking the convergence, and the empirical relationship (Eq. (14)) between the plate porosity P and porous parameter b obtained from the experiments of Cheng et al. (2016a) is applied to the following problem. Figs. 6 and 7 show the vertical displacement amplitudes at the foreend and mid-position of the VLFS as a function of non-dimensional wave numbers, respectively. Four different plate porosities, P=0.038 (b=1.673), P=0.072 (b=3.51), P=0.11 (b=5.562) and P=0.18 (b=9.343), are selected with the width 2a/L=0.04 and the submergence depth 2d 1 /L=0.025 for these perforated anti-motion plates. It can be found from both the predicted and measured values that the deflections of the VLFS is not sensitive to the plate porosity variation of the submerged plate in the long wave region kL/2<10 but becomes obviously discrepant when kL/2>10. In the aspect of design, it is to be noted that in the real configuration of a VLFS, the significant incident wave energy can at least satisfy the condition L/λ>6 (kL/2>18.84) (Watanabe et al., 2003), and thus the installation of the perforated plates attached to the edges of the VLFS is necessary for reducing the hydroelastic responses of the VLFS. In general, the perforated plates perform better when the porous parameter P   increases near to 0.11, whereas the overall performance (P=0.18) is interestingly worsened when the plate porosity further increases more than 0.11. We observe the minimum deflections of the VLFS in the case of P=0.11, which corresponds to the maximum energy dissipation. In other words, the porous parameters play a very important role in reducing the surface wave height. In these figures, the agreement between the present method and the experimental data is reasonable. The slight discrepancy between the predicted and measured data can be attributed to the possibly uncaptured nonlinear physics (such as vortices, wave breaking and splashing-over).
All the curves in Figs. 6 and 7 are given for a fixed submergence depth of anti-motion plates. To further validate the numerical model and examine the sensitivity of the position of the submerged plates on the hydroelastic responses of the VLFS, deeper submergence depth experiments of 2d 1 /L=0.0375 and 0.05 are carried out and compared with the counterpart without submerged plates. Figs. 8a-8c show the deflections at the fore-end, mid-position and back-end of the VLFS, respectively with the porosity P=0.11 (b=5.562) and the plate width 2a/L=0.04 while other parameters are   fixed by the given values. In all cases, the numerical results correlate well with the experimental results and the effectiveness of the perforated anti-motion plates in reducing the motion of the VLFS can be obviously observed. It is seen that the shallower perforated plates can more effectively reduce the hydroelastic responses of the VLFS in high frequency region kL/2>10. However, when kL/2<10, there is no appreciable improvement in the effectiveness of reducing the hydroelastic responses. Therefore, to be an effective anti-motion device, the submergence depth of the perforated plates should be smaller than 2.5% of the half-length of the VLFS for the given condition.
Then, we consider both numerically and experimentally the performance of the inclined perforated plates as the antimotion device in reducing the hydroelastic responses of the VLFS. Two different plate angles 4° and 9° with respect to the x-axis are selected and the corresponding deflections of the VLFS are shown in Figs. 9a-9c as the function of dimensionless wave numbers kL/2. Here, the porosity is P=0.11 (b=5.562) and the transverse center line of the perforated plates is clamped at 2d 1 /L=0.025. Compared with the horizontal plate (β=0°) case with the same porous parameter, the larger inclination angle slightly magnifies the vertical displacements of the VLFS in low frequency region kL/2<12.5, which is mainly due to the smaller effective plate length in the incident wave direction. However, for a moderate-length incident wave (kL/2>12.5), the deflection of the VLFS is reduced when the inclination angle is imposed smaller than 9°. Actually, the overall response-reduction performance of the VLFS can be improved by allowing small inclination angle of anti-motion plates. The experimental results follow the trend of the numerical solutions in general, and again verify the empirical assessment of the porous parameter using Eq. (14).
With the developed numerical solutions, the effect of submerged plate width on the motion of the VLFS is shown in Fig. 10 at 2d 1 /L =0.025, b 1 =5.562, β=0°, θ=0° and the particulars of the VLFS are defined in Section 4. The vertical displacement of the VLFS is plotted as a function of nondimensional wave numbers and anti-motion plate widths. As can be intuitively expected, the larger width can improve the overall effectiveness of the anti-motion perforated plates in reducing the hydroelastic responses of the VLFS. However, the construction time and economics are also necessarily considered.
Since the present theory and numerical model are introduced to cover the arbitrary incident wave angles and the real sea areas are usually multi-directional wave fields, the anti-motion effectiveness for various incident wave headings is further numerically investigated. In Fig. 11, the predicted sensitivity of oblique incident waves in reducing the hydroelastic responses of the VLFS is examined at b=5.562, 2d 1 /L=0.025, 2a/L=0.04 and β=0°. As it can be seen from these figures, there is no significant difference in the short-  wave regime (kL/2>14) but the performance of the perforated anti-motion plates is significantly improved with the larger incident wave angle when kL/2<14, which is mainly due to the changes of effective wavelengths in the direction normal to the anti-motion device i.e. larger incident angle means smaller effective incident wavelength in the x direction and thus it can more effectively reduce the motion of the VLFS for relatively longer waves.
To obtain the optimal design of the perforated anti-motion plates, extensive parametric studies have been conducted by means of the developed numerical implementation and a series of 2D contour figures for various parameters such as wave number, porosity, submergence depth, plate width, inclination angle and incident wave headings, are plotted in Figs. 12-16. By considering three positions: foreend, mid-position and back-end of the VLFS, it can be clearly observed from Figs. 12-15 that the changes of these parameters in low frequency domain kL/2<8 do not contribute much to the overall response-reduction effectivenss of the VLFS. For relatively short-wave regime (kL/2>8), it is seen from Fig. 12 that the optimal porosity for the responsereduction of the VLFS is near P=0.09-0.1, which was also confirmed in the previous experimental resluts (see Figs. 6-7). As can be seen in Fig. 13, there is also an optimal submergence depth of anti-motion plates near 2d 1 /L=0.02 when the other parameters are fixed. If the submergence 2d 1 /L ex-   ceeds 0.03, the performance of the perforated anti-motion device becomes disadvantageous with 2d 1 /L. Figs. 14a-14c show that as the width of the perforated plates increase to near 2a/L=0.04, the vertical deflections of the VLFS is reduced in the high frequency region (kL/2>10). However, when the plate width 2a/L is larger than 0.04, the deflections near the fore-end of the VLFS is magnified. Thus, the case of 2a/L=0.04 is already the optimal design and there is no effective improvement near the fore-end of the VLFS by further increasing the plate width. In Figs. 15a-15c, more calculations are presented in order to better find the optimal inclination angles of the anti-motion plates. The figure shows that the optimal range of the inclination angle in mit-igating the deflections of the VLFS is near β=8°-10°. The performance of the anti-motion device becomes worse when the inclined angle β is larger than 10° . Finally, Figs. 16a-16c show the predicted deflections of the VLFS as the function of dimensionless wavenumber kL/2 and incident wave angle θ. As is verified in our previous discussion (Fig.  11), the perforated anti-motion plates show better performance with larger oblique angle especially for relatively longer waves owing to smaller effective wavelengths in the direction normal to the anti-motion plate.

Concluding remarks
We have presented a robust numerical tool to investig-   ate the interaction of oblique regular incident waves with the VLFS attached with a pair of inclined perforated antimotion plates in the direct coupling manner for both plate and fluid part. The structural motion for the VLFS is solved by the FEM using NC-QS Mindlin plate elements. For the fluid modeling, the total potential-based BEM is employed and directly coupled with the structural formula, in which the viscous effects of the submerged plates are taken into consideration through the Darcy's law. Also, a series of experiments have been carried out by the authors in a wave basin at Dalian University of Technology to confirm the hybrid FE-BE solutions with various designed parameters. The measured results are correlated reasonably with the predicted values. Through a series of parametric studies using the developed numerical program, it can be found that there exists an optimal design for the perforated anti-motion plates in reducing the deflections of the VLFS. As a result of the parametric discussions, we observe the optimal combination of the plate porosity, submergence depth and inclination angle is near P=0.09-0.1, 2d 1 /L=0.02 and β=8°-10°, which is also supported by our measured values. The plate width 2a/L=0.04 is selected as the optimal width. After this point, the performance of the anti-motion device near the fore-end of the VLFS becomes less effective but the deflections near the mid-position and back-end of the VLFS are mitigated. The present anti-device shows better efficiency for the oblique incident waves, especially in the long wave regime, which is due to the change of effective wavelengths in the direction normal to the anti-motion plate. From the presented results, it can be concluded that the inclined perforated plates can be employed as a very effective anti-motion device attached to both the front edge and the rear edge of the VLFS and the effectiveness in reducing the hydroelastic responses of the VLFS can be enhanced by selecting the comprehensive optimal parameters.