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Advances in the Development of an Unstructured FEM Solver for Fluid-Structure Interaction Problems in Marine Engineering J. García-Espinosa, A. Coll, E. Oñate and R. Ribó, D. Sa, C. García, O. Casals, C. Corte, K. Lam International Center for Numerical Methods in Engineering (CIMNE) Compass Ingeniería y Sistemas SA (CompassIS)

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**Outline Overlapping domain decomposition level set algorithm**

Theoretical backgroung ALE-Navier Stokes iterative-implicit solver Monophase flow adaptation Application example Boundary layer resolution Anisotropic boundary layer mesh generation Stabilization aspects Fluid flow validation tests Fluid Structure interaction Development of a new generation GUI Coupling / interpolation algorithm / tools

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**Overlapping domain decomposition level set**

ODDLS theoretical backgroung ALE-Navier Stokes iterative-implicit solver Monophase flow adaptation

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**Introduction Ω1: Fluid 1 Ω2: Fluid 2**

The main problem of the free surface capturing algorithms is the treatment of the pressure (gradient) discontinuity at the interface due to the variation of the properties there. In most of these algorithms the sharp transition of the properties at the interface is smoothed (extended to a band of several elements width) and therefore the accuracy in capturing the free surface is reduced. Ω1: Fluid 1 Ω2: Fluid 2

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Introduction A new free surface capturing approach able to overcome/improve most of the difficulties/ shortcomings of the existing algorithms is presented. This approach is based on: Stabilisation of governing equations (incompressible two-fluids Navier-Stokes equations) by means of the FIC method. Application of ALE techniques. Free surface movement is solved using a level set approach. Monolithic Fractional-Step type Navier-Stokes integration scheme. Application of domain decomposition techniques to improve the accuracy in the solution of governing equations in the interface between both fluids.

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**Problem Statement: Two fluids Navier Stokes equations**

Two (incompressible) fluids (non homogeneous) Navier Stokes equations: With: And the necessary initial and boundary conditions

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**Problem statement: Level set equation**

Let be Ψ a function (level set), defined as follows: Therefore we can re-write the density field as: And finally, obtain an equivalent equation for the mass conservation in terms of Ψ (level set):

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**Problem Statement: Level set – Navier Stokes equations**

Two (incompressible) fluids level-set type Navier Stokes equations: With: And the necessary initial and boundary conditions

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**Problem Statement: ALE* formulation**

We may easily re-write the previous equations in an Arbitrary Lagrangian-Eulerian frame: Where um is the (mesh) deformation velocity of the moving domain: * The algorithm is Lagrangian for the movement of the ship but the free surface problem (level set) is solved in an Eulerian way

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**Overlapping Domain Decomposition Tecnique**

Let K be a finite element partition of domain Ω, and consider a domain decomposition of Ω into three disjoint sub domains Ω3(t), Ω3(t) and Ω5(t): Ω1 (Fluid 1) Ω2 (Fluid 2)

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**Overlapping Domain Decomposition Tecnique**

From this partition let us define two overlapping domains Ω*1, Ω*2 in such a way that Ω*1 G*2 G*1 Ω*2

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**Overlapping Domain Decomposition Tecnique**

We can write an equivalent (continuous) problem, using a standard Dirichlet-Neumann domain decomposition technique (using Ω*1, Ω*2 decomposition). The resulting variational problem is (including FIC stabilisation terms): Ω*1 Ω*2

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**Problem statement: Boundary condition at the interface**

Dirichlet conditions (compatibility of velocities at the interface) are applied on Γ*1 (boundary of And Neumann condition are applied on Γ*2 (jump condition) t* is evaluated from the resulting velocity and pressure field on Ω*1. Pressure evaluation must take into account the jump condition given by: Where γ is the coefficient of surface tension, and p1, p2 are the pressure values evaluated on the real free surface interface Γ. These are extrapolated to Γ* to impose the conditions on the defined interfaces.

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**Monolithic Fractional-Step type scheme**

Integration of Navier-Stokes equations in every domain is done by means of a Monolithic Fractional-Step type scheme. This scheme is based on the iterative solution of the momentum equation, where the pressure is updated by using the solution of a velocity divergence free correction (m iteration counter): This scheme only requires to solve scalar problems, with the subsequent savings on CPU time and memory. The discretization of the problem is done using standard FEM technique

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**Adaptation for solving Monophase Flow**

In many cases of interest in naval/marine applications, the aerodynamics effects can be neglected (density and viscosity ratio are about 1000). It is important for these cases to adapt the ODDLS technique to solve monophase problems, reducing the computational cost and capturing the free surface with the necessary accuracy and maintaining the advantages of the proposed method. In these cases, the computational domain is reduced to the nodes in the water plus those in the air being connected to the interface (Ω*1). The later nodes are used to impose the pressure and velocity boundary conditions on the interface.

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**Monophase flow algorithm**

Solve momentum conservation equation on Ω*1 using the following free surface boundary condition for the velocity: Solve the mass continuity equation on Ω*1, applying the following free surface boundary condition: Solve level set equation on Ω (update free surface position) and reinitiate distance field Evaluate forces on ship boundary and move it (update the mesh accordingly) Go to 1 until convergence.

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**Naval hydrodynamics: application example**

PT-215 Mega-yacht (Isonaval)

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Isonaval PT-215 This example shows the application of the presented technique to the analysis of a 135’ mega-yatch designed by Isonaval. The general characteristics of this yacht are shown in the following table: Main Characteristics Length overall 135’ (41,2 m) Moulded Draft 2.48 m Moulded Beam 9.5 m Construction depth 4.5 m Design Speed 14.5 Kn

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Isonaval PT-215 analyses The analyses consisted of the towing of the hull at different speeds (still water) at real scale. Analyses were carried out in parallel to experimental test in towing tank. The objective of this parallel work was to complete the information from the towink tank (such as real scale behaviour, streamlines for bilge keels arrangement, local phenomena, …) Characteristics of calculations: 2 sets of analyses were performed: fixed ship and free to sink and trim. 5 different speeds were run for every set of analysis with an unstructured 2.4 million linear tetrahedra mesh. Two more meshes of 3.7 and 6.3 million linear tetrahedra were used to study the influence of the mesh density in the results (run for the fixed ship cases). The model geometry includes keel and bow truster tunnel, but no other appendages. All the cases were run using an ILES-type turbulence model.

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**Isonaval PT-215 Streamlines (V = 14.5 kn):**

Mesh of 2.4 million tetrahedra Pressure map on hull and (limiting) streamlines

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Isonaval PT-215b After the study of the PT-215 hull, a new requirements from the owner arose: extend the LOA from 135’ to 148’. Therefore, the hull was extended amidships and a stern platform was modified and slighly enlarged. In parallel, the platform base was modified, including a slight wedge to reduce the dynamic trim angle. The final general characteristics are shown in the following table: Main Characteristics Length overall 148’ (45.1 m) Moulded Draft 2.45 m Moulded Beam 9.5 m Construction depth 4.5 m Design Speed 14.5 Kn

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**Isonaval PT-215b analyses**

The analyses consist of the towing of the hull at different speeds (still water). No parallel experimental study were done this time! Characteristics of the analyses: 1 set of analyses were performed, leaving the ship free to sink and trim. 5 different speeds (12 kn, 13 kn, 14 kn, 14.5 kn and 15 kn) were run for every set of analysis with an unstructured 2.5 million linear tetrahedra mesh. Two more meshes of 4.5 and 6.4 million linear tetrahedra were used to study the influence of the mesh density in the results (run for the full set of speed cases). The model geometry includes keel and bow truster tunnel, but no other appendages. All the cases were run using an ILES-type turbulence model.

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**Isonaval PT-215 Snapshot of the results (V = 14.5 kn):**

Mesh of 4.5 million tetrahedra Left: (dynamic) pressure map on free surface Down: pressure field on hull The results showed that the contractual speed could be achieved without increasing the power!

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**Boundary layer resolution**

Anisotropic boundary layer mesh generation Stabilization aspects Fluid flow validation tests

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**Resolution of the boundary layer**

Objective: Improvement of the resolution of the boundary layer, by generating an adapted (anisotropic) mesh in the boundary layer area. The existing (frontal method) mesh generator for the generation of unstructured tetrahedral meshes of the GiD pre-processor has been adapted. The algorithm implemented in this work is able to generate “a posteriori” boundary layer mesh of prisms or tetrahedra. The philosophy of the method is based on Y. Ito and K. Nakahashi (2002) work, but adapted to the specific requirements of the GiD system.

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**Meshing the boundary layer: steps**

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**Meshing the boundary layer: steps**

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**Meshing the boundary layer: steps**

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**Boundary layer mesh: stabilization aspects**

Finite Calculus (FIC) method is used to stabilize Navier Stokes equations. The stabilized FIC form of the governing differential equations is as follows: Characteristics lengths hm, hd and hψ must reflect the anisotropy of the mesh, since they are related to the discrete balance domain size.

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**Boundary layer mesh: Flow in a Pipe**

First validation: Turbulent flow in a pipe First element thickness 2.3·10-5 m Ratio boundary layer thickness / D = 0.004 Ratio first element thickness / D = 0.001 Turbulence model: k-ε low-Re

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**Boundary layer mesh: KVLCC2**

Flow about KVLCC2 tanker ship (validation study based on data available at Gothemburg 2000 Workshop website) Wind tunnel tests were carried out for a model at scale 1:116, resulting in a model of length 2.76 m. The velocity of the air at the inlet is 25 m/s, being the density and viscosity ρ=1.01 Kg/m³ and μ=3.045·10-5 Kg/ms, respectively. Estimated viscous sub-layer thickness 4.0·10-5 m

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**Boundary layer mesh: KVLCC2**

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**Boundary layer mesh: KVLCC2**

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**Boundary layer mesh: KVLCC2**

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Boundary layer mesh Study at real scale (L = 320 m), Re = 2.03·109. Estimated viscous sub-layer thickness is 3.0 e-5 m k-ε low-Re, Spalart Allmaras and ILES models predicted too low Cf values. k-ε low Rea and Spalart Allmaras seems to be overdiffusive and mesh density is too coarse for ILES approach. k-ω SST seems to behave better that others in this case. However it is noticeable that k-ω gives a closer value to Cf friction line for the case at scale 1/116. Similar results have been obtained in other cases (in particular NT-130 case shown afterwards).

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**Naval hydrodynamics: application examples**

NT-130 semi-planning’craft (Navtec)

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NT-130 We will apply the presented technique to the analysis of a semi-planning hull. The general characteristics of this boat are shown next. Main Characteristics LOA 14.0 m Moulded Draft 1.05 m Moulded Beam 3.54 m Design Speed 14 Kn (Fn = 0.65)

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**NT-130 Still Water Analyses**

The first example consist of the towing of the hull at different speeds (still water). Characteristics of the analyses: 2 sets of analysis were done: fixed ship and free to sink and trim. 4 different speeds were run. All the cases were run twice, using different meshes of 2.0 and 3.2 million linear tetrahedra (isotropic) and 3.4 million linear tetrahedra (anisotropic). Additional cases with different (isotropic) meshes ranging from 1.5 to 16 million linear tetrahedra were used to study the influence of the mesh density in the results.

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**NT-130 Still Water Analyses: Dynamic Sinkage and Trim effect**

Fn=0.65 Fn=0.55

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**NT-130 Still water Drag / Towing force**

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**NT-130 Still Water Analyses**

Snapshot of the results (V = 12kn): Mesh of 16 million tetrahedra Left: (dynamic) pressure Down: velocity modulus

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**NT-130 Results / Mesh dependency**

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**Fluid Structure Interaction**

Development of a new generation GUI Coupling / interpolation algorithm (library)

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**Development of a new GUI for FSI Main Objectives**

To develop a framework to integrate different analysis solvers Able to manage in an efficient way geometry and data for FSI and multi-physics solvers User-friendly and easy to use (as much as possible). Utilities to guide the user in the insertion of the data (wizards, links between data, error reporting, simplification of data tree, …) Allowing fully integration of fluid, structure and multi-physics solvers in the same GUI To develop a strategy (and tools) to minimize the effort required for the integration / communication of different (existant) solvers Resulting in a package of high quality (focussed in commercialization) Other aspects: Tools to help in the definition of materials, properties, … Parametric modeling tools Reporting tools (quasi-automatic) Library to create self-training tools Translation management tools

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**Development of a new GUI for FSI**

A new toolkit (Tcl-Tk) for managing CAE information has been developed (for GiD pre/postprocessing system) CAE Information is stored in a XML database XML is endorsed by software market leaders, it supports Unicode and it is a W3C recommendation. The syntax rules are very simple, logical, concise, easy to use. The XML documents are human-legible, clear and easy to create. The information is stored in plain text format. It can be viewed in all major of browsers and it is designed to be self-descriptive. The elements in a XML document form a tree-structure that starts at “the root” and branches to “the leaves” with different relationships between the nested elements. It allows to efficiently aggregate elements. XML elements are defined using generic tags and can have attributes, which provide additional information about elements. The new Toolkit takes advantage of this hierarchical XML structure to convert automatically the XML database to a physical tree on the GiD window. Data tree and menus are simplified/organized based on previous configuration / selection done by the user Geometry data is organized in groups and/or layers. The system allows easy and efficient integration of data for different solvers. A library for communication / interpolation of solvers based on TCL-IP sockets has been developed

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**Tree structure generated from XML database**

XML-based GUI XML-file contains Definition of data entries (general data, conditions, materials, …) TCL procedures that execute commands when necessary Xpath instructions Tree structure generated from XML database

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Toolkit for CAE GUI Start data window allows to select the main characteristics of the analysis. The XML-based data tree is adapted accordingly. Tree is dynamicaly adapted based on the data already inserted by the user.

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**Communication/Interpolation library**

A C++ library has been developed for communication of different solvers (C and FORTRAN interface). Information is interchanged at memory level, based on TCL_IP sockets. Different solvers may be connected in parallel. Library incorporates automatic mesh interpolation tools (advanced element octtree search algorithm) and allows mesh updating. Require minimum adaptation of the solvers (dynamic library).

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**FSI algorithm Weak / Iterative coupling algorithm**

FSI comm library Read new mesh A Update Mesh B Transfer fluid forces to mesh B Update mesh Solve fluid (time t) Update mesh Solve struct. (time t) Transfer structure velocity to mesh A Iterate to balance interaction forces (initially extrapolate) Read new mesh A Update Mesh B Transfer fluid forces to mesh B Update mesh Solve fluid (time t+dt) Update mesh Solve struct. (time t+dt) Transfer structure velocity to mesh A

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**Fluid Structure Interaction: application example**

NT-130 in head waves

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**NT-130 Head Waves Analyses**

This example consist of the FSI analysis of the hull structure at different speeds in head waves. Characteristics of the analyses: The analyses were carried out with the ship free to sink and trim. 4 different speeds were run for every set of analyses with two meshes: 2.9 and 3.4 million linear tetrahedra mesh (fluid). The structure of the ship was modelled using and six nodes (composite) shell triangles and beam elements. The analysed wave length range from 1.0 to 1.5 x LOA (about critical values for slamming).

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NT-130 Wave Generator The waves are generated by defining an oscillating velocity boundary condition at the inlet of the basin. The velocity is obtained from the equivalent movement of a wall given by x=d·sin(ωt), where ω is the wave generator frequency and d is the amplitude of the movement (stroke). The resulting boundary condition is as follows (k is the wave number and Vo is the ship speed): For this problem linear wave theory states that the relation between wave number, channel depth and wave-maker frequency is ω2=g·k·tanh(kh). Where k is the wave number, h is the water depth and g the gravity acceleration The waves are dumped at the right hand side of the tank by disposing a (numerical) beach.

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**NT-130 Head waves analysis**

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**NT-130 Head Waves Analyses**

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**NT-130 Head Waves Fluid dynamics analysis (right)**

Structural response (down)

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**NT-130 Head Waves analyses**

View of the mesh (longitudinal cut, free surface and ship) for a specific time step.

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Conclusions The present work describes the ODDLS methodology for the analysis of naval problems. The method is based on the domain decomposition technique combined with the Level Set technique and a FIC stabilized FEM. The ODDLS approximation increases the accuracy of the free surface capturing as well as the solution of the governing equations in the interface between two fluids. ODDLS method has also been integrated with an ALE algorithm for the treatment of the dynamic movement of a body/bodies (ship). The boundary layer treatment is based on the generation of an anisotropic mesh. The work has included the development of a FSI framework (GUI+comm libs for GiD) for integration of existing solvers. Different examples have shown practical application of the methodology.

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**For further information**

Acknowledgments This work has been partially supported by CENIT BAIP 2020 Isonaval S.A. (Spain): PT-215 analyses Navtec Ltd. (Chile): NT-130 analyses For further information

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