Elsevier

Applied Mathematical Modelling

Volume 112, December 2022, Pages 560-613
Applied Mathematical Modelling

A 3D SPH-based entirely Lagrangian meshfree hydroelastic FSI solver for anisotropic composite structures

https://doi.org/10.1016/j.apm.2022.07.031Get rights and content

Highlights

  • The first 3D SPH-based hydroelastic FSI solver for anisotropic composite structures.

  • The first hamiltonian SPH structure model for anisotropic materials.

  • Second-order consistent schemes are applied to enhance the HSPH structure model.

  • No artificial stabilizers/smoothing schemes for HSPH or ISPH-HSPH FSI solver.

  • Coherent validations for anisotropic HSPH structure model and 3D ISPH-HSPH FSI solver.

Abstract

This paper presents the first 3D entirely Lagrangian meshfree hydroelastic FSI (Fluid-Structure Interaction) solver for reproduction of incompressible fluid flows interacting with anisotropic/isotropic composite elastic structures as well as the first Hamiltonian SPH for anisotropic structures. To achieve this development, we have carefully (i) reformulated the HSPH (Hamiltonian Smoothed Particle Hydrodynamics) isotropic structure model with consideration of material anisotropy of structures, (ii) extended the 2D HSPH structure model and corresponding ISPH-HSPH FSI solver for 3D composite structures and their interactions with incompressible fluids. Regarding the advancement (i), the reformulation from isotropic to anisotropic structure model has been conducted through a careful revisit on the basis of stress-strain responses. The fourth-order elasticity tensor and transformation (rotation) of coordinate systems are considered for development of the anisotropic HSPH structure model. Then, the 3D HSPH structure model for anisotropic/composite structures is coupled with a refined projection-based Incompressible SPH (ISPH) fluid model. The proposed structure model and FSI solver are capable of handling large material anisotropies and discontinuities at material interfaces without use of any artificial stabilizers/smoothing schemes. Validations are conducted coherently. First, the newly proposed anisotropic HSPH structure model is verified through both 2D/3D classical benchmark tests with exact theoretical solutions. Then, followed by validations of HSPH for 3D composites, the corresponding coupled ISPH-HSPH FSI solver is applied to two hydroelastic FSI tests including slamming of an anisotropic composite hull.

Introduction

Anisotropic composite materials have been receiving interests in the field of structural design. In general, “anisotropic material” refers to a material that possesses different elastic moduli in different directions and distinct stress-strain responses depending on directions (Sadd [1]). The term “composite material” implies that the material is composed of two or more physically/chemically different substances with distinct material interfaces (Vasiliev and Morozov [2]). Anisotropic composite materials are designed to possess enhanced mechanical properties with respect to conventional materials and have been utilized in various engineering purposes (e.g. designing footbridge, aircraft, automotive, submarine, etc.; Rajak et al. [3]), and accordingly, anisotropic composite materials and their interaction with fluid flows have been studied by a number of researchers (e.g. Butler et al. [4], Shafei et al. [5], Kim et al. [6], Schwartzentruber et al. [7]).

Anisotropic materials and composite materials are classified into several groups. On one hand, anisotropic materials are divided into two categories (Sadd [1]): (i) naturally occurring anisotropic material (e.g. wood, rock or crystalline solid) and (ii) engineering composite materials (e.g. fibre reinforced plastic). On the other hand, composite materials are categorized into three groups (Reddy [8]): (I) laminated composites (two or more bonded layers of different materials); (II) particulate composites (macro size particle reinforced materials in a matrix); and (III) fibrous composites (fibre reinforced materials in a matrix). From a macroscopic viewpoint, considering their small-scale periodic heterogeneous feature of composites, (II) particulate and (III) fibrous composite materials can be assumed as homogeneous anisotropic materials (known as homogenization method; e.g. de Buhan et al. [9]). Whereas, (I) laminated composites have been usually analysed as heterogeneous/discontinuous composites in many numerical/theoretical studies due to their clear and simple material interfaces (e.g. Nazargah et al. [10], Khdeir and Aldraihem [11], Khandelwal et al. [12]). Reflecting on increasing applications of the laminated composites employing anisotropic materials as layer constituents (referred to as “anisotropic composite structure”, in this study), heterogeneous approach with consideration of anisotropy of each layer material is significantly important for physically consistent modelling of laminated composite material.

In some engineering fields such as ocean/coastal engineering, the FSI (Fluid-Structure Interaction) phenomena corresponding to violent free surface flows and anisotropic composite materials are often encountered in structure designs, e.g. offshore platforms, ships, hydrofoils, etc. (Mouritz et al. [13], Setvati et al. [14]). In such cases, precise modelling of overall FSI system would be quite challenging on account of the presences of strong/abrupt impact loads and structural responses, highly-deformed boundaries of fluid's free surface/fluid-structure interface, discontinuities/anisotropies of structural material properties. Accordingly, development of robust and reliable computational solvers for hydroelastic FSI corresponding to incompressible fluid flow and anisotropic composite materials would be of substantial importance.

In light of the aforementioned challenges of fluid-anisotropic composite structure interactions, entirely Lagrangian meshfree methods (Gotoh and Khayyer [15,16]) present a great potential for accurate numerical modelling of such complicated hydroelastic FSI phenomena, owing to their robustness in handling complex/large boundary deformations and violent free-surface fluid flows (e.g. Joubert et al., Tang et al., Xue et al., Tsurudome et al., Serroukh et al., Tsuruta et al., Harada et al., Tazaki et al. [17], [18], [19], [20], [21], [22], [23], [24]). Recently, considerable interest has been devoted to the theme of hydroelastic FSI solvers corresponding to free-surface fluid flows and homogeneous isotropic structures by coupling a Lagrangian meshfree fluid model either with a mesh-based structure model (e.g. Fourey et al., Zheng et al., Long et al. [25], [26], [27]) or with a particle-based structure model (leading to an entirely Lagrangian meshfree hydroelastic FSI solver, e.g. Khayyer et al., Sun et al., Zhang et al., Khayyer et al., O'Connor and Rogers, Zhang et al. [28], [29], [30], [31], [32], [33]).

Entirely Lagrangian meshfree hydroelastic FSI solvers are developed within a purely particle-based computational framework for the overall FSI systems, which can assure robust and consistent numerical modelling of complex FSI phenomena. In this context, the authors (Khayyer et al. [34]) pioneered the development of an entirely Lagrangian meshfree solver for hydroelastic FSI corresponding to composite structures. In specific, an SPH (Smoothed Particle Hydrodynamics; Gingold and Monaghan [35])-based hydroelastic FSI solver was presented through coupling of an ISPH (Incompressible SPH) fluid model with an HSPH (Hamiltonian SPH) structure model, resulting in the ISPH-HSPH hydroelastic FSI solver. The ISPH-HSPH model that was presented, however, was a 2D solver and more importantly, applicable only in case of isotropic structures. This solver was applied to a practical slamming test corresponding to anisotropic composite sandwich hull (Das and Batra, Qin and Batra [36,37]) with the anisotropic face sheet being assumed as isotropic due to limitations of the structure model in treating anisotropic materials.

Accurate numerical modelling of anisotropic structures and their corresponding FSI phenomena would be of great significance for reliable engineering structure designs. In the context of mesh-based methods, great efforts have been devoted to numerical modelling of FSI phenomena with anisotropic materials, such as interactions between gas explosions and anisotropic structures (e.g., Li et al. [38], Chiquito et al. [39]), blood flow and stenotic arteries/heart valves (e.g., Valencia and Baeza [40], Wu et al. [41]) as well as hydroelastic responses of anisotropic composite structures (e.g., Liao et al. [42], Akcabay and Young [43]). On the other hand, in the context of particle methods, there has been no research on development of entirely Lagrangian hydroelastic FSI solvers comprising anisotropic structures as well as very few studies on development of particle-based structure models for anisotropic materials. There have been a few studies on particle-based modelling of anisotropic materials in the context of discrete mechanics by DEM (Discrete Element Method), e.g., Liu and Liu [44], as highlighted in Owen et al. [45]. However, in the context of continuum mechanics, there has been no studies on particle-based modelling of anisotropic materials and its corresponding hydroelastic FSI, to our best knowledge. In addition, three-dimensional entirely Lagrangian meshfree hydroelastic FSI solvers for fluid flow interactions with laminated composite materials have not yet been developed. Certainly, such developments need to be accompanied by coherent, comprehensive and rigorous validations.

This paper sets to develop the first three-dimensional hydroelastic FSI solver for incompressible fluid flows interacting with anisotropic composite elastic structures. To achieve this development, we have carefully (i) reformulated the HSPH isotropic structure model (Khayyer et al. [34]) with consideration of structural material anisotropy, (ii) extended our recent Hamiltonian SPH (HSPH) structure model proposed for 2D composite structures (Khayyer et al. [34]) to three-dimensions. Regarding the advancement (i), the reformulation of structure model from isotropic to anisotropic has been conducted through a careful revisit on the basis of stress-strain responses (Sadd [1]). The fourth-order elasticity tensor and transformation (rotation) of coordinate systems are considered for development of the HSPH anisotropic structure model. The developed 3D HSPH structure model for anisotropic/composite structures does not include any artificial numerical stabilizers or artificial smoothing schemes, even in presence of large discontinuities in physical quantities at material interfaces and large anisotropies of materials. To our best knowledge, the paper presents the first 3D entirely Lagrangian meshfree hydroelastic FSI solver for incompressible fluid flows interacting with anisotropic composite structures, the first Hamiltonian SPH model for anisotropic structures and the first 3D Hamiltonian SPH for composite structures. For enhancement of accuracy and consistency, a second-order consistent HSPH structure model is also presented through consideration of high-order terms in Taylor-series expansion.

Validations are conducted systematically and coherently. First, the newly proposed anisotropic HSPH structure model is verified through both 2D/3D classical benchmark tests with theoretical solutions. Through the validations, the improved performance of the second-order HSPH structure model is also presented with respect to the conventional first-order HSPH one. Then, the coupled 3D ISPH-HSPH hydroelastic FSI solver is verified through conducting a FSI benchmark test with theoretical solutions corresponding to a hydrostatic water column on a composite elastic plate. Finally, the proposed FSI solver is applied, in both two and three-dimensions, to a hydroelastic slamming test corresponding to an anisotropic composite hull (Das and Batra, Qin and Batra [36,37]), where the corresponding theoretical solutions are based on the assumptions of Kirchhoff plate theory for thin face sheet, {3,2}-order sandwich composite panel theory for core and potential flow theory for fluid.

This paper comprises 5 sections and 2 appendices. Section 2 describes the basic framework of the ISPH-HSPH hydroelastic FSI solver. Section 3 presents the main novel features and developments corresponding to this paper, namely, i) anisotropic HSPH structure model and ii) 3D structure model/FSI solver for anisotropic/isotropic composite materials. A second-order accurate HSPH structure model is also presented in this section. Section 4 is devoted to validations of the proposed HSPH structure model and its corresponding coupled ISPH-HSPH FSI capable of treating anisotropic composite materials. Section 5 presents the concluding remarks of this study. In addition, the appendix sections (Appendix. A. Beam/plate theories, Appendix. B. Theoretical solutions for proposed benchmark test cases) present detailed derivations of theoretical solutions for the considered benchmark tests corresponding to anisotropic composite elastic structures.

Section snippets

Enhanced ISPH fluid model

The considered governing equations of fluid, correspond to incompressible continuity and Navier-Stokes equations:·uF=0DuFDt=pρF+νF2uF+g+aSF where D/Dt signifies Lagrangian time derivative; the quantities u, t, p, ρ, ν and g respectively refer to velocity vector, time, pressure, density, laminar kinematic viscosity and gravitational acceleration vector; the superscript F stands for the quantities of fluid particles; and aSF represents the acceleration imposed on a target fluid particle (F)

Extensions of HSPH structure model

In this section, three developments for the HSPH structure model are presented: (i) extension towards modelling of anisotropic elastic materials (Section 3.1), (ii) enhancement of consistency by considering high-order differential operators (Section 3.2), and (iii) extension for three-dimensional laminated composite materials (Section 3.3). In Section 3.1, starting with introduction of the anisotropic materials (Sadd [1], Kuna [61], Nakasone [62]), numerical modelling of anisotropic materials

Numerical validations and investigations

In this section, the accuracy, stability, convergence and conservation properties of the proposed computational methods are verified. In Section 4.1, the newly developed HSPH structure model for anisotropic materials is first verified through both 2D/3D anisotropic structure benchmark test cases, namely, 2D rotating orthotropic disc (Section 4.1.1) (Sladek et al. [74], Zhang et al. [75]), 2D rotating functionally graded polar orthotropic disc with a central hole (Section 4.1.2) (Peng and Li [76]

Concluding remarks

The present work pioneers the development of a 3D entirely Lagrangian meshfree hydroelastic FSI solver for simulation of incompressible fluid flows interacting with anisotropic/isotropic composite elastic structures. For this purpose, the so-called HSPH (Hamiltonian SPH) structure model for 2D isotropic composite structures (Khayyer et al. [34]) is further extended, enhanced and validated through (i) consideration of material anisotropy and (ii) extension of the solver to three-dimensions.

The

Acknowledgements

This study has been supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grants Number JP18K04368, JP18H03796, JP21K14250 and JP21H01433.

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