3D fluid–structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach

Finite deformation contact of flexible solids embedded in fluid flows occurs in a wide range of engineering scenarios. We propose a novel three-dimensional finite element approach in order to tackle this problem class. The proposed method consists of a dual mortar contact formulation, which is algorithmically integrated into an eXtended finite element method (XFEM) fluid–structure interaction approach. The combined XFEM fluid–structure-contact interaction method (FSCI) allows to compute contact of arbitrarily moving and deforming structures embedded in an arbitrary flow field. In this paper, the fluid is described by instationary incompressible Navier–Stokes equations. An exact fluid–structure interface representation permits to capture flow patterns around contacting structures very accurately as well as to simulate dry contact between structures. No restrictions arise for the structural and the contact formulation. We derive a linearized monolithic system of equations, which contains the fluid formulation, the structural formulation, the contact formulation as well as the coupling conditions at the fluid–structure interface. The linearized system may be solved either by partitioned or by monolithic fluid–structure coupling algorithms. Two numerical examples are presented to illustrate the capability of the proposed fluid–structure-contact interaction approach.     

Imposing rigidity constraints on immersed objects in unsteady fluid flows

Imposing rigidity constraints of an immersed elastic body in a transient flow field is not trivial. It requires solution stability and accuracy. In this paper, we present an efficient and accurate algorithm implemented to enforce fluid–structure interface constraints used in the immersed finite element method (IFEM). This interface treatment is a constraint applied onto the rigid bodies based on the fluid structure interaction force evaluated from the immersed solid object. It requires no ad hoc constants or adjustments, thus providing numerical stability and avoiding unnecessary trial-and-error procedures in defining the stiffness of the elastic body. This force term can be evaluated for both uniform and nonuniform fluid grids based on the higher order interpolation function adopted in the IFEM. The ability in handling nonuniform interpolations offers the convenience in modeling arbitrary geometrical shapes and provides solution refinements around interfaces. The results we obtained from flow past a rigid cylinder demonstrate that this convenient way of constraining the interface is a reliable and robust numerical approach to solve unsteady fluid flow interacting with immersed rigid bodies.     

Synchronized reproducing kernel interpolant via multiple wavelet expansion

In this paper, a new partition of unity – the synchronized reproducing kernel (SRK) interpolant – is derived. It is a class of meshless shape functions that exhibit synchronized convergence phenomenon: the convergence rate of the interpolation error of the higher order derivatives of the shape function can be tuned to be that of the shape function itself. This newly designed synchronized reproducing kernel interpolant is constructed as an series expansion of a scaling function kernel and the associated wavelet functions. These wavelet functions are constructed in a reproducing procedure, simultaneously with the scaling function kernel, by directly enforcing certain orders of vanishing moment conditions. To the authors knowledge, this unique interpolant is the first of its kind to be constructed, and to be used in numerical computations, both in concept and in practice. The new interpolants are in fact a group of special hierarchial meshless bases, and similar counterparts may exist in spline interpolation method, other meshless methods, Galerkin-wavelet method, as well as the finite element method. A detailed account of the subject is presented, and the mathematical principle behind the construction procedure is further elaborated. Another important discovery of this study is that the 1st order wavelet together with the scaling function kernel can be used as a weighting function in Petrov-Galerkin procedures to provide a stable numerical computation in some pathological problems. Benchmark problems in advection-diffusion problems, and Stokes flow problem are solved by using the synchronized reproducing kernel interpolant as the weighting function. Reasonably good results have been obtained. This may open the door for designing well behaved Galerkin procedures for numerical computations in various constrained media.     

Three-dimensional modal analysis of an idealized human head including fluid–structure interaction effects

A three-dimensional analytical and numerical method is presented in this article for the analysis of the acoustic fluid–structure interaction systems including, but not limited to, the brain, cerebro-spinal fluid (CSF), and skull. The model considers a three-dimensional acoustic fluid medium interacting with two solid domains. This article deals with the analytical and numerical computation of eigenproperties for an idealized human head model including fluid–structure interaction phenomena. We determine in the present work the natural frequencies and the modes shapes of the system of the brain, cerebro-spinal fluid (CSF), and skull. Two models are presented in this study: an elastic skull model and a rigid model. In the analysis, a potential technique is used to obtain in three-dimensional cylindrical coordinates a general solution for a solid problem. A finite element method analysis is also used to check the validity of the present method. The results from the proposed method are in good agreement with numerical solutions. The effects of the fluid thickness and compressibility on the natural frequencies are also investigated.     

Semi-implicit formulation of the immersed finite element method

The immersed finite element method (IFEM) is a novel numerical approach to solve fluid–structure interaction types of problems that utilizes non-conforming meshing concept. The fluid and the solid domains are represented independently. The original algorithm of the IFEM follows the interpolation process as illustrated in the original immersed boundary method where the fluid velocity and the interaction force are explicitly coupled. However, the original approach presents many numerical difficulties when the fluid and solid physical properties have large mismatches, such as when the density difference is large and when the solid is a very stiff material. Both situations will lead to divergent or unstable solutions if not handled properly. In this paper, we develop a semi-implicit formulation of the IFEM algorithm so that several terms of the interfacial forces are implicitly evaluated without going through the force distribution process. Based on the 2-D and 3-D examples that we study in this paper, we show that the semi-implicit approach is robust and is capable of handling these highly discontinuous physical properties quite well without any numerical difficulties.     

Mathematical foundations of the immersed finite element method

In this paper, we propose an immersed solid system (ISS) method to efficiently treat the fluid–structure interaction (FSI) problems. Augmenting a fluid in the moving solid domain, we introduce a volumetric force to obtain the correct dynamics for both the fluid and the structure. We further define an Euler–Lagrange mapping to describe the motion of the immersed solid. A weak formulation (WF) is then constructed and shown to be equivalent to both the FSI and the ISS under certain regularity assumptions. The weak formulation (WF) may be computed numerically by an implicit algorithm with the finite element method, and the streamline upwind/Petrov–Galerkin method. Compared with the successful immersed boundary method (IBM) by Peskin and co-workers (J Comput Phys 160:705–719, 2000; Acta Numerica 11:479–517, 2002; SIAM J Sci Stat Comput 13(6):1361–1376, 1992) the ISS method applies to more general geometries with non-uniform grids and avoids the inaccuracy in approximating the Dirac delta function     

Treatment of discontinuity in the reproducing kernel element method

A discontinuous reproducing kernel element approximation is proposed in the case where weak discontinuity exists over an interface in the physical domain. The proposed method can effectively take care of the discontinuity of the derivative by truncating the window function and global partition polynomials. This new approximation keeps the advantage of both finite element methods and meshfree methods as in the reproducing kernel element method. The approximation has the interpolation property if the support of the window function is contained in the union of the elements associated with the corresponding node; therefore, the continuity of the primitive variables at nodes on the interface is ensured. Furthermore, it is smooth on each subregion (or each material) separated by the interface. The major advantage of the method is its simplicity in implementation and it is computationally efficient compared to other methods treating discontinuity. The convergence of the numerical solution is validated through calculations of some material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

LLM and X-FEM based interface modeling of fluid–thin structure interactions on a non-interface-fitted mesh

This paper presents a non-interface-fitted mesh method for fluid–thin structure interactions. The key components are the Lagrangian Lagrange-multiplier (LLM) method and the extended finite element method (X-FEM). The LLM couples fluid and thin structure through the Lagrangian nodes of the structure element. The X-FEM gives flow discontinuity to the fluid elements intersected by the structure element. The combination method is verified through applications to flow with a domain-partitioning boundary and flow-induced flapping of a flexible filament. We discuss how the discontinuities at the interface enhance the simulation results, how the lack of the discontinuities affects the results, and identify some effects of these discontinuity enrichments.     

Numerical modeling of complex interactions between underwater shocks and composite structures

To study the complex interactions between underwater shocks and composite structures, a strongly coupled Eulerian–Lagrangian numerical solver is developed. The coupled numerical solver consists of an Eulerian fluid solver, a Lagrangian solid solver, a one-fluid cavitation model, and an interface capturing scheme. The interface capturing scheme features a fluid characteristics method and a modified ghost fluid method (MGFM). The MGFM is reformulated for fluid–solid coupling by treating simultaneously the fluid characteristics equation and the solid equation of motion to determine the interface variables, leading to a strongly coupled Eulerian–Lagrangian scheme. Various components of the numerical solver are first individually tested and validated. The strongly coupled solver is then applied to realistic shock-structure interaction problems involving composite structures. The accuracy of the coupled solver is demonstrated via comparison with numerical predictions and experimental observations available in literature. Finally, the validated coupled numerical solver is utilized to study the effectiveness of a proof-of-concept shock mitigation scheme.     

A reproducing kernel method with nodal interpolation property

A general formulation for developing reproducing kernel (RK) interpolation is presented. This is based on the coupling of a primitive function and an enrichment function. The primitive function introduces discrete Kronecker delta properties, while the enrichment function constitutes reproducing conditions. A necessary condition for obtaining a RK interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points. A normalized kernel function with relative small support is employed as the primitive function. This approach does not employ a finite element shape function and therefore the interpolation function can be arbitrarily smooth. To maintain the convergence properties of the original RK approximation, a mixed interpolation is introduced. A rigorous error analysis is provided for the proposed method. Optimal order error estimates are shown for the meshfree interpolation in any Sobolev norms. Optimal order convergence is maintained when the proposed method is employed to solve one‐dimensional boundary value problems. Numerical experiments are done demonstrating the theoretical error estimates. The performance of the method is illustrated in several sample problems. Copyright © 2003 John Wiley & Sons, Ltd.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

On the coupling between fluid flow and mesh motion in the modelling of fluid–structure interaction

Partitioned Newton type solution strategies for the strongly coupled system of equations arising in the computational modelling of fluid–solid interaction require the evaluation of various coupling terms. An essential part of all ALE type solution strategies is the fluid mesh motion. In this paper, we investigate the effect of the terms which couple the fluid flow with the fluid mesh motion on the convergence behaviour of the overall solution procedure. We show that the computational efficiency of the simulation of many fluid–solid interaction processes, including fluid flow through flexible pipes, can be increased significantly if some of these coupling terms are calculated exactly.     

Computation of free-surface flows and fluid–object interactions with the CIP method based on adaptive meshless soroban grids

The CIP Method [J comput phys 61:261–268 1985; J comput phys 70:355–372, 1987; Comput phys commun 66:219–232 1991; J comput phys 169:556–593, 2001] and adaptive Soroban grid [J comput phys 194:57–77, 2004] are combined for computation of three- dimensional fluid–object and fluid–structure interactions, while maintaining high-order accuracy. For the robust computation of free-surface and multi-fluid flows, we adopt the CCUP method [Phys Soc Japan J 60:2105–2108 1991]. In most of the earlier computations, the CCUP method was used with a staggered-grid approach. Here, because of the meshless nature of the Soroban grid, we use the CCUP method with a collocated-grid approach. We propose an algorithm that is stable, robust and accurate even with such collocated grids. By adopting the CIP interpolation, the accuracy is largely enhanced compared to linear interpolation. Although this grid system is unstructured, it still has a very simple data structure.     

Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods

The stabilized space–time fluid–structure interaction (SSTFSI) technique developed by the Team for Advanced Flow Simulation and Modeling (T★AFSM) was applied to a number of 3D examples, including arterial fluid mechanics and parachute aerodynamics. Here we focus on the interface projection techniques that were developed as supplementary methods targeting the computational challenges associated with the geometric complexities of the fluid–structure interface. Although these supplementary techniques were developed in conjunction with the SSTFSI method and in the context of air–fabric interactions, they can also be used in conjunction with other moving-mesh methods, such as the Arbitrary Lagrangian–Eulerian (ALE) method, and in the context of other classes of FSI applications. The supplementary techniques currently consist of using split nodal values for pressure at the edges of the fabric and incompatible meshes at the air–fabric interfaces, the FSI Geometric Smoothing Technique (FSI-GST), and the Homogenized Modeling of Geometric Porosity (HMGP). Using split nodal values for pressure at the edges and incompatible meshes at the interfaces stabilizes the structural response at the edges of the membrane used in modeling the fabric. With the FSI-GST, the fluid mechanics mesh is sheltered from the consequences of the geometric complexity of the structure. With the HMGP, we bypass the intractable complexities of the geometric porosity by approximating it with an “equivalent”, locally-varying fabric porosity. As test cases demonstrating how the interface projection techniques work, we compute the air–fabric interactions of windsocks, sails and ringsail parachutes.     

An investigation of Interface-GMRES(R) for fluid–structure interaction problems with flutter and divergence

The basic subiteration method for solving fluid–structure interaction problems consists of an iterative process in which the fluid and structure subsystems are alternatingly solved, subject to complementary partitions of the interface conditions. The main advantages of the subiteration method are its conceptual simplicity and its modularity. The method has several deficiencies, however, including a lack of robustness and efficiency. To bypass these deficiencies while retaining the main advantages of the method, we recently proposed the Interface-GMRES(R) solution method, which is based on the combination of subiteration with a Newton–Krylov approach, in which the Krylov space is restricted to the interface degrees-of-freedom. In the present work, we investigate the properties of the Interface-GMRES(R) method for two distinct fluid–structure interaction problems with parameter-dependent stability behaviour, viz., the beam problem and the string problem. The results demonstrate the efficiency and robustness of the Interface-GMRES(R) method.     

Topology optimization of creeping fluid flows using a Darcy–Stokes finite element

A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no‐slip condition along the solid–fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy–Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no‐slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid–fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy–Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0–1 (void–solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material. Copyright © 2005 John Wiley & Sons, Ltd.     

Wall slip induced by a micropolar fluid

Lubrication theory is devoted to the study of thin-film flows, More often, the fluid can be considered as a Newtonian one and no-slip boundary conditions can be retained for the velocity at the fluid solid interface. With these assumptions it is possible to deduce from the (Navier) Stokes system a simplified equation describing the flow: the Reynolds equation. It allows to compute the pressure distribution inside the film and to obtain overall performances of a lubricated device such as load and friction coefficient. For very thin films, however, surface effects at the fluid solid interface become very important and no-slip conditions cannot be retained. Solid surfaces exert some influence on the liquid molecules and the effective shear viscosity along the boundary differs from the classical bulk shear viscosity. Moreover, the microstructure of the fluid cannot be ignored, especially the effects of solid-particle additives in the lubricant. Micropolar theory for fluids is often adopted to account of such microstructure and microrotation. In the present study, a thin micropolar fluid model with new boundary conditions at the fluid–solid interface is considered. This condition links velocity and microrotation at the interface by introducing a so-called “boundary viscosity”. By way of asymptotic analysis, a generalized micropolar Reynolds equation is obtained. Numerical results show the influence of the new boundary conditions for the load and friction coefficients. Comparisons are made with other works that retain the no-slip boundary conditions.