Project

MarineSEM : Spectral Element Methods for Nonlinear Waves, Wave-Structure and Wave-Body modelling for Marine Hydrodynamics

Goal: Collaborative research that focus on researching, improving, developing and applying robust, fast and accurate spectral element methodologies with adaptive mesh capabilities for weakly and fully nonlinear and dispersive free surface wave modelling models, fixed and moving wave-body methods for marine offshore engineering applications.

Date: 1 January 2014

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Jens Visbech
added a research item
Due to society's present and future demand for sustainable and green energy resources, the offshore industry calls for faster and more accurate simulation tools to develop and optimize structures at sea. These tools are based on mathematical formulations and numerical discretization methods that - due to the rapid increase in computational resources - looks to become the primary and dominant go-to option when considering engineering tasks in the ocean. The main objective of this thesis has been to develop and validate a new 2D/3D numerical model that solves the linear radiation problem in terms of an impulsive and pseudo-impulsive time domain formulation. This model serves the purpose of efficiently simulating wave-induced response and loading on floating offshore structures by calculating the hydrodynamic added mass and damping coefficients. The weak Galerkin formulation of the governing potential flow equations was derived and discretized using the spectral element method to enable geometrical flexibility, efficient computational scalability, and high accuracy. An explicit 4-stage 4'th order Runge Rutta scheme performed the time integration combined with a global CFL condition on the temporal discretization. To complement the new numerical models, the thesis also included topics such as: Structured and unstructured meshes, affine and curvilinear elements, 2D and 3D mesh generation using MATLAB and Gmsh for a variety of element types, wave generation for numerical wave tanks, non-reflective and periodic domains, continuous and discrete Fourier transformation, linear wave propagation and analysis in 2D and 3D, eigenvalue stability analysis of discretizations, pressure computation using finite difference approximations, and a novelty investigation and discussion of unresolved energy in free surface quantities. To demonstrate the legitimacy of the developed models, various validation tests were carried out by comparing against known benchmarks and analytical solutions for different floating structures, such as a cylinder, a barge, a simplified spar buoy, a sphere, and a box. In general, excellent numerical results were obtained, ultimately confirming the correctness of the model implementation.
Jens Visbech
added 2 research items
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 O(N p), with N begin the total number of grid points and p ≈ 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.
Estimating the hydrodynamic characteristics of bodies interacting with ocean waves is of key importance in ocean engineering. Solving the wave-structure interaction problem has been a topic of research for many years, especially using linearized potential flow theory which generally captures the majority of the physics for typical marine structures. Linear theory allows for a decomposition of the potential into radiation and diffraction parts, where the focus of this abstract lies on the former. For modelling the linear interactions between floating bodies and ocean waves, the impulsive time-domain formulation has been widely used and studied [1], and in recent years the pseudo-impulsive approach has gained renewed attention [2]. The basic concept for these impulsive methods is to force the floating object with an impulse (or pseudo-impulse) in velocity and measure the resultant force on the body. The Fourier transform of the force divided by the Fourier transform of the body motion conveniently determines the added mass and damping coefficients for all frequencies. Despite the work by Robertson & Sherwin (1999) [3] suggesting an inherent mesh-instability problem with the spectral element method (SEM) applied to free surfaces waves, significant progress for pure wave propagation and wave-structure interaction using SEM have been made over the last half-decade starting with Engsig-Karup, Eskilsson & Bigoni (2016) [4]. Using this high-order numerical discretization method allows for exceptional geometrical flexibility, high accuracy and efficiency, and an optimal O(n) scaling of the computational effort. This latter property is achieved in combination with p-multigrid techniques [5, 6]. For a further review of the SEM and its beneficial capabilities see [7]. This abstract seeks to highlight recent progress towards developing a computational tool in the setting of linear potential flow using a pseudo-impulsive approach combined with a higher-order SEM. This ultimately enables all of the aforementioned features of this numerical method, including curvilinear elements, unstructured meshes, different radiation conditions, and much more.
Allan Peter Engsig-Karup
added 3 research items
Submitted to IWWWFB 2022. This abstract describes our recent work on employing reduced-order modelling (ROM) to solve fully nonlinear potential flow equations (FNPF) to achieve faster turn-around time than a full order model (FOM) based on the spectral element method (SEM) [1]. We propose a PODGalerkin based model-order reduction approach to reduce the cost of the solve step in the Laplace problem. If repeated simulations are needed for applications, e.g. in optimization loops with varying parameters, it may become prohibitively expensive to run many FOM simulations in practical times. Reduced-order modelling techniques were introduced to eliminate the time-consuming behaviour of high-dimensional numerical methods and reduce the load on computational resources without compromising overall accuracy. The proper orthogonal decomposition (POD) method is one of the most effective snapshot-based reduced-order modelling techniques and is considered in this work. The basic idea of using POD is to generate a low-dimensional model with few degrees of freedom using the most dominant features of the system, thereby significantly reducing the computational time and cost.
Submitted to IWWWFB 2022. Simulation of water waves and taking into account the sea floor to estimate sea states are important for the design of offshore structures. We propose a new high-order accurate pseudospectral method for solving the incompressible Navier-Stokes equations with a free surface. The work is motivated by the lack of high-order accurate free surface water wave models which include viscous and rotational effects. The numerical scheme utilizes Fourier and Chebyshev basis functions for the numerical discretization and enables exploiting the fast Fourier transform (FFT) algorithm for efficiency reasons. A key feature is an explicit-implicit pressure-correction method designed to work with general Runge-Kutta (RK) methods used for temporal integration that fulfills mass balance and a pressure-velocity coupling. Another key feature is the use of a Fourier-continuation technique for solving numerical wave tank problems on finite (nonperiodic) domains. As a starting point towards establishing the solver, we provide benchmark results to demonstrate accuracy and convergence using established cases used for potential flow solvers. We present numerical case results for i) nonlinear stream function wave solutions and ii) steep solitary wave reflection in a numerical wave tank in this abstract.
Submitted to IWWWFB 2022. See the published version. Estimating the hydrodynamic characteristics of bodies interacting with ocean waves is of key importance in ocean engineering. Solving the wave-structure interaction problem has been a topic of research for many years, especially using linearized potential flow theory which generally captures the majority of the physics for typical marine structures. Linear theory allows for a decomposition of the potential into radiation and diffraction parts, where the focus of this abstract lies on the former. For modelling the linear interactions between floating bodies and ocean waves, the impulsive time-domain formulation has been widely used and studied [1], and in recent years the pseudoimpulsive approach has gained renewed attention [2]. The basic concept for these impulsive methods is to force the floating object with an impulse (or pseudo-impulse) in velocity and measure the resultant force on the body. The Fourier transform of the force divided by the Fourier transform of the body motion conveniently determines the added mass and damping coefficients for all frequencies. Despite the work by Robertson & Sherwin (1999) [3] suggesting an inherent mesh-instability problem with the spectral element method (SEM) applied to free surfaces waves, significant progress for pure wave propagation and wave-structure interaction using SEM have been made over the last half-decade starting with Engsig-Karup, Eskilsson & Bigoni (2016) [4]. Using this high-order numerical discretization method allows for exceptional geometrical flexibility, high accuracy and efficiency, and an optimal O(n) scaling of the computational effort. This latter property is achieved in combination with p-multigrid techniques [5, 6]. For a further review of the SEM and its beneficial capabilities see [7]. This abstract seeks to highlight recent progress towards developing a computational tool in the setting of linear potential flow using a pseudo-impulsive approach combined with a higher order SEM. This ultimately enables all of the aforementioned features of this numerical method, including curvilinear elements, unstructured meshes, different radiation conditions, and much more.
Allan Peter Engsig-Karup
added a research item
We present a Spectral Element Fully Nonlinear Potential Flow (FNPF-SEM) model developed for the simulation of wave-body interactions between nonlinear free surface waves and impermeable structures. The solver is accelerated using an iterative p-multigrid algorithm. Two cases are considered: (i) a surface piercing box forced into vertical motion creating radiated waves and (ii) a rectangular box released above its equilibrium resulting in freely decaying heave motion. The FNPF-SEM model is validated by comparing the computed hydrodynamic forces against those obtained by a Navier-Stokes solver. Although not perfect agreement is observed the results are promising, a significant speedup due to the iterative algorithm is however seen.
Allan Peter Engsig-Karup
added a research item
In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave-body interaction with high computational efficiency within a single modelling approach. We design and propose an efficient O(n)-scalable computational procedure based on geometric p-multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high-order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p-multigrid spectral element model can take advantage of the high-order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h-multigrid approaches. We provide numerical benchmarks for the algorithmic and numerical efficiency of the iterative geometric p-multigrid solver. Results of numerical experiments are presented for wave propagation and for wave-body interaction in an advanced case for focusing design waves interacting with a FPSO. Our study shows, that the use of iterative geometric p-multigrid methods for the Laplace problem can significantly improve run-time efficiency of FNPF simulators.
Allan Peter Engsig-Karup
added a research item
We present a fully nonlinear potential flow (FNPF) model for simulation of wave-body interaction in three spatial dimensions (3D) and apply it to the case of an axi-symmetric point absorber. The FNPF model is discretized is space by a C 0 spectral element method (SEM) using high-order prismatic-possibly curvilinear-elements. This SEM-FNPF model is stabilized following the work presented in [2] and the wave-body interaction is solved by the acceleration potential method [4]. Following the work of [3] the model is based on an Eulerian formulation and the direct discretization of the Laplace problem makes it straightforward to handle accurately floating bodies. The FNPF-SEM approach has been illustrated to have the potential to deliver a computationally efficient tool for wave-body interaction [3]. In this work we apply the model to the 2nd test case of the OES Task 10 project: a heaving point absorber made up of surface piercing body with a cylinder on top, and a conical frustum on the bottom. This case was experimentally investigated in [1]. We present computations of diffraction, radiation and decay tests as well as heave response in regular wave. REFERENCES [1] Coe, stabilised nodal spectral element model for fully nonlinear water waves.
Allan Peter Engsig-Karup
added a research item
POST-CONFERENCE ISOPE 2018 CONTRIBUTION: A 3D fully nonlinear potential flow (FNPF) model based on an Eulerian formulation is presented. The model is discretized using high-order prismatic - possibly curvi-linear - elements using a spectral element method (SEM) that has support for adaptive unstructured meshes. The paper presents details of the FNPF-SEM development and the model is illustrated to exhibit exponential convergence for steep stream function waves to serve as validation. The model is then applied to the case of focused waves impacting on a surface-piercing fixed FPSO-like structure. Good agreement is found between numerical and experimental wave elevations and pressures. KEY WORDS: Spectral element method; high order numerical methods ; unstructured meshes; fully nonlinear potential flow; focused wave; wave-body interaction; FPSO.
Allan Peter Engsig-Karup
added a research item
The need for advanced time-domain simulators for improved offshore engineering analysis is growing with the continued improvement in computational resources. In line with this trend, we consider a fully nonlinear potential flow (FNPF) model discretised with a stabilised Galerkin Spectral Element Method (SEM) [5] addressing the stability problems and lack of progress made for this type of modelling approach since the work of Robertson & Sherwin (1999) [9]. A recent review of the SEM is given in [12] and the benefits of high-order discretizations and multigrid for FNPF models are well-known, e.g. see [1, 2]. In a recent study, a SEM-based FNPF model (FNPF-SEM) model [4] was developed and then validated in a blind test experiment against experimental measurements for focusing waves interacting with a fixed FPSO structure. In this work, we consider a standard benchmark problem for cylindrical structures due to McCamy & Fuchs (1954), with the objective of evaluating an entirely new extension of this solver with a p-multigrid method which enables scalable O(n) complexity in work effort. Recently, the first proof of an efficient geometric p-multigrid method was demonstrated in 2D/3D [7], and in this work we provide some additional results with the aim of demonstrating the practical feasibility of using this new SEM-based solver for 3D analysis. In particular, the new FNPF-SEM p-multigrid solver makes it possible to address both wave propagation and wave-structure problems within a single solver. The p-multigrid method is designed to exploit the p-type convergence property in solving the Laplace problem, and by avoiding an h-type convergence strategy, it is possible to handle the representation of structural bodies with curvilinear features without refining the underlying mesh-topology. In this sense, this work contributes to demonstrating that the SEM can be an efficient basis for a technology that is useful for engineering analysis. It comes with the ability to represent offshore structures and the discretization leads to sparse matrices after global assembly in the discrete problem, and therefore can be solved iteratively with high parallel efficiency and scalability. By using a FNPF formulation it is possible to predict the wave-induced horizontal and vertical hydrodynamic forces on offshore structures, and account for the nonlinear effects that are significant when standard frequency domain analysis falls short.
Allan Peter Engsig-Karup
added a research item
Results from Blind Test Series 1, part of the Collaborative Computational Project in Wave Structure Interaction (CCP-WSI), are presented. Participants , with a range of numerical methods, simulate blindly the interaction between a fixed structure and focused waves ranging in steepness and direction. Numerical results are compared against corresponding physical data. The predictive capability of each method is assessed based on pressure and run-up measurements. In general, all methods perform well in the cases considered, however, there is notable variation in the results (even between similar methods). Recommendations are made for appropriate considerations and analysis in future comparative studies. KEY WORDS: Code comparison; numerical validation; CFD; FNPT; PIC; hybrid codes; focused waves; range of steepness; range of incident wave angle; FPSO; run-up and pressure on bow.
Allan Peter Engsig-Karup
added a research item
Nonlinear wave-body problems are important in renewable energy, especially in case of wave energy converters operating in the near-shore region. In this paper we simulate nonlinear interaction between waves and truncated bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results if applied within the underlying assumptions of dispersion and nonlinearity but with a significant reduction in computational effort.
Allan Peter Engsig-Karup
added a research item
Breather solutions to the nonlinear Schrödinger equation have been put forward as a possible prototype for rouge waves and have been studied both experimentally and numerically. In the present study, we perform high resolution simulations of the evolution of Peregrine breathers in finite depth using a fully nonlinear potential flow spectral element model. The spectral element model can accurately handle very steep waves as illustrated by modelling solitary waves up to limiting steepness. The analytic breather solution is introduced through relaxation zones. The numerical solution obtained by the spectral element model is shown to compare in large to the analytic solution as well as to CFD simulations of a Peregrine breather in finite depth presented in literature. We present simulations of breathers over variable bathymetry and 3D simulations of a breather impinging on a mono-pile.
Allan Peter Engsig-Karup
added a research item
We present a spectral/hp element method for a depth-integrated Boussinesq model for the efficient simulation of nonlinear wave-body interaction. The model exploits a 'unified' Boussinesq framework, i.e. the flow under the body is also treated with the depth-integrated approach, initially proposed by Jiang [25] and more recently rigorously analysed by Lannes [28]. The choice of the Boussinesq equations allows the elimination of the vertical dimension, resulting in a wave-body model with an adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. In this work we employ flux-based conditions for domain coupling, following the recipes provided by the discontinuous Galerkin spectral/hp element framework. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model. The spectral/hp element method allows to achieve high-order, possibly exponential, convergence for non-breaking waves and account for the nonlinear interaction with fixed and floating bodies. Our main contribution is to include floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is validated against published results for wave-body interaction and confirmed to have excellent accuracy. The proposed nonlinear model is demonstrated to be relevant for the simulation of wave energy devices.
Allan Peter Engsig-Karup
added a research item
PRE-CONFERENCE ISOPE 2018 CONTRIBUTION: For the assessment of experimental measurements of focused wave groups impacting a surface-piecing fixed structure, we present a new Fully Nonlinear Potential Flow (FNPF) model for simulation of unsteady water waves. The FNPF model is discretized in three spatial dimensions (3D) using high-order prismatic - possibly curvilinear - elements using a spectral element method (SEM) that has support for adaptive unstructured meshes. This SEM-FNPF model is based on an Eulerian formulation and deviates from past works in that a direct discretization of the Laplace problem is used making it straightforward to handle accurately floating structural bodies of arbitrary shape. Our objectives are; i) present details of new SEM modelling developments and ii) to consider its application to address a wave-body interaction problem for nonlinear design waves and their interaction with a modelscale fixed Floating Production, Storage and Offloading vessel (FPSO). We first reproduce experimental measurements for focused design waves that represent a probably extreme wave event for a sea state represented by a wave spectrum and seek to reproduce these measurements in a numerical wave tank. The validated input signal based on measurements is then generated in a NWT setup that includes the FPSO and differences in the signal caused by nonlinear diffraction is reported.
Allan Peter Engsig-Karup
added a research item
The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.
Allan Peter Engsig-Karup
added 15 research items
The main objective of the present study has been to develop a numerical model and investigate solution techniques for solving the recently derived high-order Boussinesq equations of \cite{MBL02} in irregular domains in one and two horizontal dimensions. The Boussinesq-type methods are the simplest alternative to solving full three-dimensional wave problems by e.g. Navier-Stokes equations, which can capture all the important wave phenomena such as diffraction, refraction, nonlinear wave-wave interactions and interaction with structures. The main goal can be reached by using multi-domain methods with support for a spatial discretization based on unstructured grids. In the current work, a standard method of lines approach has been adapted, and the method of choice for the spatial discretization is the nodal Discontinuous Galerkin Finite element method (DG-FEM), which provides a highly flexible basis for the model. This method is combined with an explicit Runge-Kutta method for the temporal discretization. The resulting discrete set of equations enables us to simulate water waves accurately in complex geometric settings and possibly employ local adaption techniques to optimize the computational effort. The high-order Boussinesq equations constitute a highly complex system of coupled equations which put any numerical method to the test. The main problems that need to be overcome to solve the equations are the treatment of strongly nonlinear convection-type terms and spatially varying coefficient terms; efficient and robust solution of the resultant time-dependent linear system; and the numerical treatment of high-order and cross-differential derivatives. The suggested solution strategy of the current work is based on a collocation approach where the DG-FEM is used to approximate spatial derivatives and the boundary conditions are imposed weakly using a symmetry technique. Since collocation methods are prone to aliasing errors, various anti-aliasing strategies are applied for the stabilization of the models. A practical and relatively straightforward discretization is applied, which is based on a simple treatment of slip boundary conditions at wall surfaces. A linear Fourier analysis has been applied to obtain generic analytical results which can be used for validating the discrete implementation and provide the basis for choosing stable discretization parameters as well as giving new insight into the properties of the high-order Boussinesq equations. Remarkably, it is demonstrated that the linear eigenspectra of the linearized semi-discrete equation system is bounded and hence the stable time increment is not dictated by the spatial discretization. This is a favorable property for explicit time-integration schemes as the stable time increment is not subject to severe restrictions which can affect the performance of the scheme. It is demonstrated that the discrete properties of both DG-FEM and finite difference methods can be discretized to mimic the analytical properties. It is investigated mathematically and demonstrated numerically how the relaxation method of \cite{LD83} can be applied in spectral/$hp$ multi-domain methods for both accurate internal wave generation of arbitrary wave fields and efficient absorption near domain boundaries. The method is considered to be particular attractive for wave generation purposes for use with high-order Boussinesq models as it alleviates the need for specifying consistent boundary conditions, and importantly, it is a very straightforward and flexible method. The DG-FEM models have been applied to a number of tests in both one and two horizontal dimensions with the objective of both validating the setup against known analytical and experimental test results, and at the same time demonstrating the attractive properties of the method. It has been demonstrated that difficult nonlinear and dispersive wave problems can be solved accurately in one horizontal dimension. In two horizontal dimensions it has been demonstrated that the model can solve problems in both regular and irregular geometries and by comparison with analytical results it is shown that the results are in general in excellent agreement. Thus, it has been established that the DG-FEM can be used to solve this relatively complicated system of equations. The computational efficiency of the method has yet to be demonstrated.
We introduce a new stabilized high-order and unstructured numerical model for modeling fully nonlinear and dispersive water waves. The model is based on a nodal spectral element method of arbitrary order in space and a σ-transformed formulation due to Cai, Langtangen, Nielsen and Tveito (1998). In the present paper we use a single layer of quadratic (in 2D) and prismatic (in 3D) elements. The model has been stabilized through a combination of over-integration of the Galerkin projections and a mild modal filter. We present numerical tests of nonlinear waves serving as a proof-of-concept validation for this new high-order model. The model is shown to exhibit exponential convergence even for very steep waves and there is a good agreement to analytic and experimental data.
A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.
Allan Peter Engsig-Karup
added a project goal
Collaborative research that focus on researching, improving, developing and applying robust, fast and accurate spectral element methodologies with adaptive mesh capabilities for weakly and fully nonlinear and dispersive free surface wave modelling models, fixed and moving wave-body methods for marine offshore engineering applications.