This chapter will precisely review literature that tackles the various methods that can be used in solving FSI problems, for example, overlap mesh, semi-implicit algorithm, conforming-mesh, flux-based Lagrange multiplier, XFEM-based embedded fluid formulation, numerical simulation, a hybrid ALE-fixed-grid monolithic FSI approach, computational fluid-structure interactions, least squares, and immersed domain methods. These methods are relevant in rendering remedy to the numerous FSI problems.

2.1 Introduction

Fluid-structure interaction involves the interaction of one or more solid structure with surrounding or internal fluid flow. FSI has many areas of applications and it is a common phenomenon in nature. The fluid forces acting on the structure determine the structural movement, and since the forces are in action, the entire structure exhibits some level of motion. As noted by Chakrabarti and other authors, the problems of fluid-structure interaction do play a vital role in several engineering and scientific fields. However, studying such problems has always remained a challenge due to the multidisciplinary and strong nonlinearity nature of FSI. Obtaining an analytical solution to the majority of the FSI model equations seems cumbersome; however, experiments from the laboratory are scarce. Numerical simulations, therefore, become handy when investigating the physics taking place in most complex interactions involving solids as well as fluids. Simulations of both engineering and scientific systems of late have become complicated and sophisticated with the recent improvements in the computer technology. According to Mucha et al. other areas of applications of fluid-structure interactions include sedimentation, aerodynamics, particle assembly, ciliary beating, magneto-hydrodynamics, and complex flows in irregular domains. Two approaches may be used when classifying the numerical procedures for solving fluid-structure problems. The two approaches are partitioned and monolithic approaches.

As noted by Hubner et al., Michler et al., and Ryzhakov et al., the monolithic approach tends to consider both the structural dynamics and fluid in one similar framework of mathematics, thereby forming for the entire problem a single system equation. The equation formed is then simultaneously solved using a unified algorithm. However, in the solution procedure, the interface conditions are implicit, and this approach tends to achieve precise results for every multidisciplinary problem. The only problem with it is that it requires more resources as well as expertise in the development and maintenance of such a specialized code. On the other hand, the partitioned approach tends to consider the structure and the fluid as different computational areas capable of being solved separately with their numerical algorithm and discretization mesh. These interfacial conditions in the mesh discretization do communicate the information between structure solutions and the fluid. The partitioned approach integrates structural and fluidic discipline algorithms and performs the code development reduction through looking into the merit of the legacy code as well as numerical algorithms already validated and capable of tackling complex structural and fluids problems (Berthelsen & Faltinsen, 2008). When the partitioned method is successful, it can solve a problem of fluid-structure interactions of a structural physics and sophisticated fluid. However, partitioned approach has one challenge and the challenge is coordination of the algorithms disciplines in order to acquire a correct FSI solution with little or no modification of codes (Causin Gerbeau & Noblile, 2005).

Treatment of overlap meshes form part of the fluid-structure interaction procedures, and the two overlap methods are conforming overlap mesh and non-conforming overlap mesh methods. In the case of the conforming overlap mesh method, the interface conditions are considered as a physical boundary condition, and this condition tends to consider the interface location as a solution form. However, it needs an overlap mesh with conformity capabilities to the interface. On the contrary, the non-conforming overlap mesh methods tend to consider the interface condition as well as boundary location as constraints placed on the model equations to enable the employment of non-conforming overlap meshes. The solid and fluid interactions can be solved independently with their grids, however, re-meshing is unnecessary.

When it comes to numerical modelling, the FSI fluid’s phase can be described using Eulerian as the datum. However, for solid phase the description can be presented using a Lagrangian formulation. The immersed boundary (IB) method as proposed by Peskin is capable of solving moving boundary with the ability to simulate heart and heart valves. Other methods to solve FSI problems discussed in the paper are ALE grid approach, fixed grid approach, numerical simulation, and XFEM with Lagrange multipliers.

2.2 Overlap meshing in fluid-structure interface

Treatment of overlap meshes form part of the fluid-structure interaction procedures, and the two overlap methods are conforming overlap mesh and non-conforming overlap mesh methods. In the case of the conforming overlap mesh method, the interface conditions are considered as a physical boundary condition, and this condition tends to consider the interface location as a solution form. However, it needs an overlap mesh with conformity capabilities to the interface. On the contrary, the non-conforming overlap mesh methods tend to consider the interface condition as well as boundary location as constraints placed on the model equations to enable the employment of non-conforming overlap meshes. The solid and fluid interactions can be solved independently with their grids, however, re-meshing is unnecessary. Three fields, namely; fluid dynamics, mesh movement, and structural dynamics are described by the FSI methods with overlapping conforming meshes. The consistency existing between structural codes and fluid as well as coordination of data transfer forms the basis of emphasis of these methods. According to Antonio Huerta, generalized Gauss-Seidel (GGS) approach is used by the FSI methods for purpose of coupled analysis. In the coupled analysis, the structural and fluidic computational are performed in a sequential manner with the sole purpose of achieving a multidisciplinary solution. It involves first solving the fluid field with an assumed interface location at a given time. However, the resultant stress and pressure are then applied as external forces to the structure. The position of the structural surface is updated via conduction of the structural computation ( Huerta, A, and Calmet, H , 2014). The new interface surface is then accommodated by the newly created fluid mesh. In order to satisfy the force and displacement conditions of the interfacial conditions, an iterative process shall be employed at the very moment before moving ahead to the next instance. Maintenances of proper data transfer between the various disciplines tend to be the challenge encountered when computing using the iterative coupled procedure.

In overlapping conforming-mesh methods, the fluid dynamic module seems to closely consider the physics details found in the fluid-structure interface. The model for the fluid dynamics mesh, will always represent the structure geometry, for example, struts, hard chine, and tunnel details. On the contrary, the module for the structure analysis considers only the force bearing members. The structural mesh refinement, however, will be placed on areas with high stress, and these areas are unlikely to be on the fluid-structure interface. Gaps as well as mismatches are contained on the associated meshes found on the interface, and this only depend on the degree of fidelity employed on the structure and fluid computation. However, with the incongruence, numerical difficulties might arise when dealing with transfers of fluid dynamic load as well as elastic deformation update. Two approaches have been proposed to provide remedy for these numerical difficulties, and these approaches are artificial structure method and the point match method. When it comes to the point match method, the first step is to locate and fix the fluid mesh point on the fluid-structure interface to the structural mesh point. The displacement at the selected mesh points of the fluid surface need to be determined first and then using global and local interpolation, the displacement vectors found on the rest of the fluid surface mesh can be obtained. The pressure existing on the fluid surface mesh is the aerodynamics pressure load and this pressure is transferrable to the structure surface mesh on the condition that there is a good relation between the connection in the pair of the identify match products. The consistency of the virtual work always helps in the completion of the process. The work done on the fluid dynamic load applied to the structural surface mesh is directly proportional to the work done on the structural load applied.

2.3 Semi-implicit algorithm methods for solving problems

The literature that records about semi-implicit schemes are mainly linked or connected to monolithic methods of solving fluid structure interfaces, where in other word we say that the scheme narrows down to the implicit algorithms. When semi-implicit monolithic scheme is compared to a solution derived from dt = 10^{-6} which denotes an implicit monolithic methodology. However, according to available literature, there are a number of strategies or ways that have been brought forward by different authors whose aim is to tackle problems related to fluid-structures (Cristobal B, Alfonso Caiazzo, Miguel Angel Fern´andez, 2013). For us to understand the point here, it is very important to mention that the type of software used, creates the bases of classification, for instance when the sub problems of the structure and the fluid are tackled uniquely by a solver or rather a software then this is termed as direct or implicit method. According to Axel Gerstenberger, in his publications, he records that there is a coupling concept that he calls partially explicit scheme which is also referred to as semi-implicit scheme (Gerstenberger A, and Wall A, W, 2012). He adds that this scheme has a number of similarities to that Breuer independently proposed. In brief, Axel argues that, following the prior work of Chorin which dates back in 1968, more research have been carried out which mainly involved the flow of incompressible fluid and whose scheme is enlarged thus converging to Fluid structure interface. From their research, the projection sub step was strongly coupled with the structure and done in a fluid domain which was known which the idea is presented on partitioned schemes where we have strongly and weakly or loosely coupled schemes (Annalisa Q, and Quarteroni A,, 2011).

When considering semi-implicit scheme, we cannot fail to mention partitioned approach which tends to consider both the structure dynamics and fluid in one similar framework of mathematics thereby forming for the entire problem a single system equation. The equation formed is then simultaneously solved using a unified algorithm. However, in the solution procedure, the interface conditions are implicit or even semi-implicit, and this approach tends to achieve precise results for every multidisciplinary problem (Michael B, Guillaume D, N, Manuel M, Thomas G,, 2014). The only problem with it is that it requires more resources as well as expertise in the development and maintenance of such a specialized code. On the other hand, the partitioned approach tends to consider the structure and the fluid as different computational areas capable of being solved separately with their numerical algorithm and discretization mesh. These interfacial conditions in the mesh discretization do communicate the information between structure solutions and the fluid. The partitioned approach integrates structural and fluidic discipline algorithms and performs the code development reduction through looking into the merit of the legacy code as well as numerical algorithms already validated and capable of tackling complex structural and fluids problems. When the partitioned method is successful, it can solve a problem of fluid-structure interactions of a structural physics and sophisticated fluid (Michael B, Guillaume D, N, Manuel M, Thomas G,, 2014). However, partitioned approach has one challenge and the challenge is coordination of the algorithms disciplines in order to acquire a correct FSI solution with little or no modification of codes. When using the Yosida method or rather approach the solution derived from the scheme is compared at various time steps to that of semi-implicit or partially explicit monolithic scheme which has been computed based on the very same mesh. This Yosida method is known for its error which behaves or acts like dt^{ 2} in all quantities that are available and this leads to a conclusion that semi- implicit or partially explicit schemes that are based on Yosida method do converge linearly to implicit monolithic solution method. As a result of this explicit approach or rather semi-implicit scheme the Yosida method brings with it a computational cost reduction advantage.

Again another area that covers semi-implicit coupling is simulation of blood flow, where in such instances; the implicit coupling methods or schemes do retain energy equilibrium. According to various studies, semi-implicit coupling methods do provide the best stability properties which remains at that state for a broad range of discrete and physical parameters. The primary idea, that makes this scheme effective, is the issue of implicit coupling of the structure with pressure stress, thus making the assembly stable and at the same time slash down the computational cost.

2.4 Conforming-mesh methods

Again, the three fields, namely; fluid dynamics, mesh movement, and structural dynamics are portrayed by the FSI systems with acclimating lattices. The consistency existing between auxiliary codes and fluid and also coordination of information exchange frames the premise of accentuation of these techniques. As indicated by Newman et al., summed up Gauss-Seidel (GGS) methodology is utilized by the FSI strategies with the end goal of coupled investigation. In the coupled examination, the basic and fluidic processing is performed in a consecutive way with the sole reason for accomplishing a multidisciplinary arrangement. It includes first tackling the fluid field with an expected interface area at a given time. In any case, the resultant push and weight are then connected as outside strengths to the structure. The position of the auxiliary surface is redesigned through conduction of the basic calculation. The new interface surface is then suited by the recently made liquid cross section. Keeping in mind the end goal to fulfil the power and removal states of the interfacial conditions, an iterative procedure should be utilized at the exact second before making headway to the following occurrence. Systems for upkeeps of legitimate information exchange between the different orders have a tendency to be the test experienced when registering utilizing the iterative coupled technique.

2.4.1 Interface data transfer

Although in the conforming-mesh methods, the fluid element module appears to firmly consider the material science points of interest found in the fluid structure interface. The model for the fluid element lattice, will dependably speak to the structure geometry, for instance, struts, hard chine, and passage points of interest (Shyy, Udarkumar, Rao & Smith, 2007). Unexpectedly, the module for the structure examination considers just the power bearing individuals. The auxiliary cross section refinement, in any case, will be set on regions with high stretch, and these ranges are unrealistic to be on the liquid structure interface. Crevices and jumbles are contained on the related cross sections found at the interface, and this just relies on upon the level of constancy utilized in the structure and liquid reckoning. In any case, with the incongruence, numerical challenges may emerge when managing exchanges of fluid element stack and additionally flexible distortion overhaul. Two methodologies have been proposed to give a solution for these numerical challenges, and these methodologies are counterfeit structure system and the point match strategy. Concerning the point match system, the first step is to find and fix the liquid lattice point on the liquid structure interface to the auxiliary cross section point. As per Berthelsen and Faltinsen (2008), the auxiliary or liquid lattice focuses under determination may be vertex, the Gauss point or the focal point of a component of a cross section. Berthelsen and Faltinsen (2008) went further to say that the interrelationship between the focuses under coordinating may be set up through determination of the briefest separation inside of the focuses. Then again, this separation can likewise be acquired through the typical projection as noted by (Onishi et al, 2008). Utilizing an inflexible component fit for interfacing the coordinated focuses, the exchange of the dislodged auxiliary lattice point to the fluid surface may be conceivable. The uprooting of the chose cross section purposes of the fluid surface should be resolved first and afterward utilizing worldwide and nearby introduction, the dislodging vectors found in whatever remains of the fluid surface lattice can be gotten. The weight existing on the fluid surface lattice is the study of air weight burden and this weight is transferable to the structure surface work on the condition that there is a decent connection between the association between the pair of the distinguish match items. The consistency of the virtual work dependably helps in the consummation of the procedure. The work done on the liquid element load connected to the basic surface cross section is straightforwardly relative to the work done on the auxiliary burden connected. The moderate part of the heap regarding exchanging system does not ensure the methodology. Cebral and Lohner (2007) both built up a heap projection technique which is progressive, and the strategy should exchange the liquid element load (Cebra & Lohner, 2007). An exceptional association technique was produced by Samareh (2009), and the strategy was equipped for thinking seriously about the state of the outline representation. As per the recommendations given by Samareh (2009), a non-uniform normal B-spline (NURBS) representation was initially made so as to help with displaying the flying machine wing (Samareh, 2009). Nonetheless, on the structure surface cross section focuses, the auxiliary removals were never exchanged specifically to the coordinated fluid surface lattice focuses. Rather, they were anticipated on the NURBS model, and a representation of the disfigured geometry is developed utilizing NURBS, and this gave a foundation of the new liquid element surface.

2.4.2 Enforcement of interface constraints

In a much earlier work, Dolbow and Franca noted that weak constraints are applied in the fixed-grid methods; however, they are present during the applications conditions in the mesh free methods, though these mesh free methods tend to lack the interpolation property in the finite element. Four principle approaches are used when applying for constraints more so to the continuous Galerkin. The analysis of the approaches is emphasized in order to determine their applicability to the NS 3-D equations. For purposes of the incompressible NS equations, the first and fourth approaches are implemented. According to Dolbow and Franca, introduction of recent approaches based on residual free bubbles has not been reviewed (Dolbow, J, & Harari I,, 2009; Dolbow, J and Franca L, 2008). The governing equation in its strongest form includes the interface condition. Enforcement of interface constraints tends to formulate the coupling between the velocity on the interface mesh and the fluid velocity of the background grid. In a three-dimensional case, the interface mesh contains quadrilateral and triangular elements. However, the implicit surface of the physical fluid found in the background mesh lacks such surface elements. Through the interpolation via hexahedral and tetrahedral domain elements, the stress and velocity along the surface for the background mesh can be obtained. The definition of two sets is capable of highlighting the surface coupling character. All the DOFs from intersected element nodes are contained in set C, whereas the remaining pressure DOFs and nodal velocity are contained in set S. Similarly, Forster and Ramm present a Nitsche-XFEM technique for liquid structure interaction issues, including a flimsy walled versatile structure (Lagrangian formalism) drenched in an incompressible thick liquid (Eulerian formalism) (Forster W, & Ramm E,, 2007). The liquid area is discretized with an unstructured work not fitted to the strong mid-surface cross section. Frail and solid discontinuities over the interface are taken into account the speed and weight, individually. The liquid strong coupling is implemented reliably utilizing a variation of Nitsche’s technique with cut-components. Strength concerning discretionary interface convergences is ensured through suitable adjustment.

In other studies, Heil, Hazel, and Boyle considered a communication issue in a liquid structure in which the liquid is depicted by the Stokes comparisons and the structure by a direct thin film or shell model (Heil, M., Hazel, A, & Boyle, J., 2008). The liquid area is signified by _ Rd (d = 2; 3). The structure is thought to be inundated inside, with its mid-surface, spoke to by the situated complex _ of co-dimension 1 and unitary typical vector n. According to Fries and Zilian, the fluid structure disintegrates into two open spaces 1 and 2 (Fries, T and Zilian, A., 2009).

By iterations, the result becomes complemented with standard initial conditions u (0) = u0, d(0) = d0 and .d(0) = .d0. Here, the constant f and s stand for the fluid and solid densities, respectively, while denoting the solid thickness. The fluid Cauchy-stress tensor is given by (u; p) def = pI + 2(u) with (u) def = ru +ruT and denoting the fluid dynamic viscosity. The abstract surface differential operator L describes the solid elastic effects.

In earlier studies conducted by Oden and Reddy, the Hellinger-Reissner principle is a blend or two-field arrange for that displays the additional flux q¯ as a self-sufficient and twofold field in the fundamental temperature field c (Oden & Reddy 4). The bar over the flux exhibits that the flux is as of now a free field as opposed to the flux field q that is a part of the temperature. Moreover, the variational utilitarian of the Hellinger-Reissner vocational standard for the far reaching evacuation issue of a thin, shallow shell with an optional shape is at first settled. By then the helpfulness of the modified rule suitable for the constrained part procedure is induced. In the functional only two free variables, the shirking wand the uneasiness limit F is included. Moreover, perfect duality comes as a result of the *generalized Hellinger*–*Reissner principle (Gao 4)*. The evacuation expressions in the middle surface on the farthest point of the shell are moreover controlled by strategy for as far back as two variables.

Flux and temperature are discretized self-governing such that must be weakly maintained. The discrete q¯h is approximated element-wise fitful and can be merged at the part level. The immediate structure before development is of the structure.

One of the crucial uses of the HR system is the advancement of low-demand parts without locking (Lee, Y., Yoon, K & Lee, P., 2012). For a survey on such multi-variable specifying for solid mechanics, the important segment of the present issue is that after development, interface conditions are managed sadly without additional Lagrange multiplier inquiries. A use of the HR procedure to weak interface conditions in the setting of XFEM has been presented, starting late in (Fries, T and Zilian, A., 2009). The drawback is that the part strength system is modified for all or if nothing else for each and every met segment. That would inescapably oblige an absolutely new part anticipate the normal application on the offset NS numerical articulations.

Moreover, the Hellinger-Reissner is composed of elements, which sets a fantastic standard in the class of blended HR components. By and by, there were a few fruitful endeavours in the writing to acquire better execution. On account of the same objective, we grow in this paper four-hub blended components utilizing a basic methodology: the established and the incremental HR utilitarian are utilized either as a part of the first shape or in the improved structure; 5- and 7-parameter stress representations are expected in different directions, including the skew, regular and Cartesian coordinates (Turska, E., & Wisniewski, K,., 2008).

Thusly, we have the capacity to add to two components that perform indistinguishably as the most precise components, which, on the other hand, are substantially more confounded and utilize more parameters (Suri, M, 2005). These components are more powerful shaping twists for coarse cross sections than the PS component. Especially, basic and productive is our HR5-S component taking into account the non-upgraded HR functions, and the 5-parameter stress representation in skew directions. It utilizes a much less number of parameters than the components, yet it gives the same precision. The created components are taking into account the Green strain and consolidate the incremental constitutive methodology representing the plane anxiety condition. They are indicated to function admirably for huge deformity issues and non-straight materials.

2.4.2.1 Flux-based Lagrange multiplier

According to studies conducted by Brable, lagrange multipliers are utilized if one wants to solve obliged optimized problems. That is, assume you have a capacity, say *f* (x, y), for which you need to discover the most extreme or least esteem (Brable, J, 2009). In any case, you are not permitted to consider all (x, y) while you search for this value. Rather, the (x, y) you can consider are obliged to lie on some bend alternately surface. There are loads of samples of this in science, designing and financial aspects, for illustration, upgrading some utility capacity under spending plan limitations. Lagrange multipliers problem is presented as:

Minimize (or expand) w = f (x, y, z) obliged by g (x, y, z) = c (Jamshidi 148).

On the other hand, lagrange multipliers solution is represented as:

Nearby minima (or maxima) must happen at a basic point. This is a point where Vf = λVg, what’s more, g(x, y, z) = c (Jeremy 1).

Similarly, it is important to know that there are good ways of finding extrema even though, in some cases, there is no need for explicit formula (Trench,W, 2013). If the first essential in the equation is supplanted with its exceptional casing, which is independently in perspective of the temperature and its test limit, then the going too feeble structure is obtained Again, temperature c and flux q¯ are free variables and the temperature slant in light of each vital variable is coupled sadly. Regardless, the flux enters the equation fair by method from the point of confinement fundamental, where Dirichlet conditions are actualized weakly. The standard blend by parts of the first crucial in the equation gives from which the Euler-Lagrange numerical articulations are recovered. The temperature shape limits Nc and I (x) is picked as piecewise reliable, polynomials, that are co-reliable at between part confines. For the XFEM approach, both temperature and flux field are irregular over the interface and both are enhanced along Gi. Moreover, Nitsche’s strategy has as of late increased extraordinary consideration in the connection of certain interfaces demonstrated by the XFEM (Dolbow, J, & Harari I,, 2009), additionally as a broad system for the burden of imperatives along non-coordinating surface networks (Bazileys, Y., and Hughes, T, 2008).

For the examples, temperature and flux have the same solicitation for their shape limits. Getting the documentation from, the unpleasant estimate for temperature and flux could be termed as Q1Q−1 for trilinear, Q, Q (20) −2 for quadratic fortunes and Q2Q−2 for full triquadratic shape limits. In addition, the appraisal for immediate and quadratic tetrahedrons can be described as P1P−1 and P2P−2, independently. Other part shapes like pyramids and wedges should vague the flux in a firmly taking after way. Note that an investigative proof for robustness is up ’till now missing. Within the present work, simply a numerical contribution with the new approach and the first HR framework gives sureness that modification is excessive.

The upper right-corner contains simpler utmost terms. The remaining terms are the same as in the HR-definition. This specifying decouples the flux scientific explanation from the crucial weak warmth conduction correlation, if no restriction crosses the part. Hence, the standard (or one-field) FE itemizing is genuine for all non-crossed segments; meanwhile, each and every focalized segment has flux inquiries described on the part. Due to the element wise-irregular flux close estimation, these part fluxes can be thick on the segment level and a changed segment immovability system is made with just temperature addresses. The lattice C_{cc} is generally disproportionate. The strayed method for a sooner or later back symmetric issue may be seen as a drawback, regardless, for the arranged application – the NS numerical articulation with its convection term – this property does not confound the game plan of the discrete system.

Approaches for prerequisites in uncontrollable Galerkin (DG) schedules have not been considered in purpose of enthusiasm, in light of the way that the subject of this proposition is a CG technique for the NS numerical articulation. Eventually, the investigation performed on irregular Galerkin (DG) frameworks may upgrade the understanding of CG methods. For example, Lew and Buscaglia remark that either the crucial variable c or the twofold variable l should be unpredictable. While the DG procedure cuts down the movement essential for the fundamental variable, the proposed system cuts down the congruity of the Lagrange multiplier variable q¯. We can also take a gander at the Lagrangian as an encoding of the issue. This perspective is straightforward (yet doesn’t generally go anyplace) (Benson, D, 2010). At whatever point the imperatives are fulfilled, *g _{i} *are zero, thus on these points, paying little heed to the estimation of the multipliers.

You could envision utilizing the Lagrangian to do obliged augmentation in the accompanying way. You move *x* around *R ^{n}* searching for a more extreme estimation of

*f*. Nonetheless, you have no power over, which gets set in the most exceedingly bad routes feasible for you. Thusly, when you pick

*x, lambda*is decided to minimize. This is a decent certainty to remember. You could envision utilizing the Lagrangian to do obliged amplification in the accompanying way. Be that as it may, you have no influence over, which gets set in the most noticeably bad routes workable for you. Consequently, when you pick is decided to minimize. Formally, the issue is to discover which gives, and presently recollect that if your x happens to fulfil the imperatives.

The issue with the above perspective of the Lagrangian is that it truly doesn’t finish anything past encoding the requirements and giving us back the same issue we began with: locate the greatest benefit of, overlooking the estimations of which are not in the plausible locale. This is a piece of the full Kuhn-Tucker hypothesis, which we are not going to demonstrate thoroughly, in any case, the instinct behind why it’s actually been critical. Before we inspect why this inversion ought to work, how about we see what it fulfils in the event that it is valid. We initially had a compelled advancement issue. We would all that much like for it to turn into an unconstrained enhancement issue. It may not be clear why that is any diverse that settling and discovering a minimizing worth. The Lagrange multiplier technique likewise covers the instance of imbalance requirements (Klein). The requirements of this structure are composed *h (x) *≥ 0. The key perception about imbalance imperatives work is that, at any given *x, *a *h* (*x*) can be correct either. In the event that œ8‑ then œ is said to be dynamic at *h *(*x*), else it is dormant. On the off chance that h (*x*) = 0 is dynamic at *x*, then is a great deal like a fairness imperative; it permits *x *to be greater if the angle of ∆*f *(*x*), is either zero or guiding towards negative estimations of (which damage the requirement). Be that as it may, if the slope is guiding towards positive estimations of *h*, then there is no reason that we cannot move for each.

2.4.3 Fluid-structure interface conditions

Two conditions, namely kinematic and dynamic coupling conditions are the main conditions present at the interface (Degroote, B,. et al, 2008). Assumptions of impermeable structure surfaces are carried out such that across the interface, there is no occurrence of mass flow. Matching of the normal velocities at the interface is necessary, and the normal vectors do have an opposite sign since the normal vectors are different for both structural domain and fluid (Beale & Layton, 2006). Taking into considerations the viscous fluids, a matching condition is usual for the tangential velocities. The tangential velocities can be combined with normal vector equation to obtain the condition of no slip boundary (Sohn et al, 2010; Vierendeels, Dumont & Verdonck, 2008). There is a requirement by the force equilibrium for equality with surface traction, however, the position of the force equilibrium keeps on varying with time and through the interaction of both fields it can be defined (Shen & Chan, 2008).

2.5 Numerical analysis

It is noted that an original SGI 3400 machine contains 32 processors and all these processors are based on scalable distributed-shared-memory silicon graphics. Parallel computations are carried out on this SGI 3400 machine, and the machine has a multi-processing architecture (Tai, C, H,. Zhao, Y, & LiewK, M,., 2005). Apart from the processors, the machine contains 8 nodes with each node having 4 MIPs 64 bits. St. Jude Medical (SJM) with an opening of 29mm together with MHV was studied numerically with the already mentioned methods. In order to carry out this study, a set of time-dependent inlet velocity equation needs to be devised, and the equations have their basis on the physiological mass flow rate similar to the normal human rate. The OG mesh has the capacity to combine both overlapping and background meshes. However, for purposes of downstream effect elimination, the distance between the valve and the outlet for the mesh needs to be at least 10 times the flow channel diameter. The location of the valve and the inlet should be 75 mm, however, for the outlet the distance needs to 275 mm (Sanders et al, 2010). The performance of grid convergence on various meshes is considered to be optimum. In order to compute the flow for the background mesh, a three-level MG needs to be employed (Tai, C, H,. Zhao, Y, & LiewK, M,., 2005). For the background flow domain, the valve leaflet should be immersed and positioned at 60 degrees in order to define the physical boundary of the immersed leaflet (Wall, A, et al, 2008).

2.5.1 Numerical simulation

The numerical simulation methods, also known as Newton-Raphson method can suitable tackle the FSI problems and it is employed in both partitioned and monolithic approaches. The method is capable of solving non-linear flow equations as well as the structural equations in the entire solid and fluid domain. The Newton-Raphson iteration has a system of linear equations which can be solved entirely without the cooperation of Jacobian though it requires a matrix-free iterative method. When a matrix free iterative method is employed, a finite difference approximation of the product vector of Jacobian-vector needs to be in place. Even though the numerical simulation solves both structures and the flow problem, formulation of the FSI problem with degrees of freedom needs to be in place in the interface position. Decomposition of the domain condenses FSI problem errors into related interface subspace allowing the problem to be written as a fixed point or root finding problem. This root-finding problem can be solved by numerical simulation with Newton-Raphson iterations, for example, approximation of Jacobian a linear model. However, this technique employs the interface block quasi-Newton technique, and it approximates least-squares Jacobian models that in turn formulate the FSI problem into an equation though stress distribution and the interface position remains unknown. Fixed-point iterations solve fixed-point problems and these problems are also known as Gauss-Seidel iterations meaning the structure and flow problem are successively solved giving rise to a smaller change as compared to the convergence criterion.

2.4 An XFEM-based embedded fluid formulation

Finite deformation contact with flexible solids implanted in liquid streams happens in an extensive variety of designing situations. Henke proposes a novel three-dimensional Finite component approach with a specific end goal to handle this issue class (Henke, 2013). The proposed strategy comprises of a double mortar contact definition, which is algorithmically incorporated into an eXtended Finite component system (XFEM) fluid–structure association approach. The joined XFEM fluid–structure-contact interaction strategy (FSCI) permits to process contact of discretionarily moving and disfiguring structures implanted in a discretionary stream field. In his paper, the liquid is portrayed by in stationary incompressible Navier–Stokes comparisons. A definite fluid–structure interface representation grants to catch stream designs around reaching structures precisely and in addition to mimic dry contact between structures. No confinements emerge from the basic and the contact plan. Dolbow and Franca determined a linearized solid arrangement of mathematical statements, which contains the liquid detailing, the auxiliary plan, the contact definition and in addition the coupling conditions at the fluid–structure interface (Dolbow, J and Franca L, 2008). The linearized framework may be fathomed either by apportioned or by solid, fluid–structure coupling calculations.

Two numerical samples are displayed to outline the capacity of the proposed liquid structure-contact interaction approach. Flimsy walled, structures are in view of the Arbitrary Lagrangian Eulerian (ALE) strategy. These methodologies backtrack to ahead of schedule works like. The fundamental element of ALE based routines is that the liquid field is defined and comprehended on a disfiguring framework (Wall, A & Hansbo, P., 2010). This lattice distorts with the structure at the interface and afterward the matrix deformation is stretched out into the liquid field. Yet, even the most progressive and best refined ALE based plan once goes as far as possible where just re-cross section makes a difference. At the most recent in such circumstances, one may be enticed to swing over to methodologies that work with an altered matrix (Tai, C, H,. Zhao, Y, & LiewK, M,., 2005). Here, the interface is depicted either unequivocally, utilizing a Lagrangian interface, markers or a Lagrangian basic discretization, or verifiable, utilizing e.g. Level-set capacities on an altered liquid lattice. Changing properties and discontinuities in the liquid arrangement must be dealt with adjustments on the liquid mathematical statements and/or liquid discretization.

Conspicuous altered matrix systems for incompressible stream incorporate the Immersed Boundary (IB) strategy and its numerous inductions. It is competent to recreate flimsy and deformable limits and completely fledged, deformable 3d structures submersed in incompressible stream. A methodology with numerous likenesses to the IB strategy is the supposed Distributed Lagrange Multiplier/ Fictitious Domain (DLM/FD) system (Forster, Wall, A & Ramm E., 2009). Initially, the methodology was produced for unbending particles with translational and rotational degrees of flexibility. The DLM/FD systems have following been reached out to reenact meager, deformable basic surfaces and additionally to flexible and completely fledged structures. Both routines have in like manner a Lagrangian network for the structure proceeding onward top of the liquid work and compelling the liquid material to disfigure as the structure (Souli & Benson, 2010). Without being finished, underneath. Gro and Reusken gives a rundown of weaknesses of which no less than one applies to each of the current techniques and for which require upgrades before altered framework systems can turn out to be as boundless as ALE based FSI calculations.

Future routines ought to permit a free work size for each of the mimicked fields. Additionally, there ought to be no impediment on how thin structures can be as for the liquid field framework size. Liquid structure, communication is of extraordinary pertinence in numerous fields of designing also is in the connected sciences (Mayer, A & Wall, A, 2009)(Mayer and Wall 850). Consequently, the improvement and utilization of particular re-enactment methodologies has increased extraordinary consideration over the previous decades. Some present trials in this field are: the headway from uncommon reason or unique issue to very broad methodologies; the craving to try and catch extremely broad and complex frameworks; and the urgent need of vigorous superb methodologies notwithstanding for such complex cases, i.e. Approaches that can possibly turn over from being a testing and an entrancing exploration theme to an improved instrument with genuine prescient capacities. Regularly, when associated impacts are key this joins extensive basic deformations. Nevertheless, numerous accessible methodologies (both in examination and additionally in business codes) need strength particularly in this circumstance.

This is because of a representation of the general issues of the interaction of a stream field and a flexible structure, for this situation an implanted structure. The conjoined interface FSI isolates the auxiliary spaces from the liquid area f. Most research and business codes accessible for reproductions of the communication of streams and flexible, regularly. As of not long ago, none (to the learning from the creators) of the right now accessible settled matrix systems meets the high necessities expressed previously. Among others, these necessities must be satisfied to match or surpass the ALE approach as for relevance, precision and numerical solidness. They serve us as a rule for the advancement of another settled matrix approach that guarantees to conquer the tended to inadequacies.

While trying to meet these necessities, (Ember A, Dolbow, J & Harari, I., 2010)Embar, Dolbow, and Harari propose an iterative coupling plan between a standard Lagrangian basic depiction and an Eulerian definition for the liquid that uses components of the eXtended Finite Element Method (XFEM) and the DLM/FD systems specified previously (Embar, Dolbow & Harari 880). The XFEM is utilized to appropriately depict the interface, including discontinuities of the kinematic variables and in the force equalization (Gamntzer, P & Wall, A, 2006). The XFEM was initially presented for the reproduction of Cracks and different discontinuities in structures and have been, near to the current subject, reached out into issues of two-stage stream and Stokes stream/unbending molecule interaction. Gro and Reusken embraced this plan and show how the solid calculation in can be changed for an iterative coupling and how cumbersome and slim structures can be effortlessly mimicked (Gro, S & Reusken, A, 2007). Moreover, the utilization of extra level-sets to depict the interface as utilized as a part of could be stayed away from.

With this XFEM based approach, in vital the greater part of the said deficiency can be tended to, most unmistakably, there is no impact on the invented liquid area any longer and, for an adequate substantial imaginary liquid space; a critical number of pointless liquid questions can be evacuated. The interface can speak to the best possible discontinuities, despite the fact that the exactness and the cross sectional size reliance in the middle of the liquid and basic discretization needs promote studies.

2.5. A hybrid ALE-fixed-grid monolithic FSI approach

In research and commercial, the most widely and popularly used approaches are ALE (Arbitrary Lagrangian-Eulerian) based FSI methods, where the flow field formulation is derived from ALE formulation, and the structure is explained based on Lagrangian framework. The flow field is solved on a deformable grid as it is allowed by the fluid field’s ALE formulation based on a researched principle which states that at a common fluid structure interface, both the structure and the deformable grid do undergo a similar deformation (Francis C, and Radovitzky, 2005). The fluid mesh deformation within the domain are unrestrained, and as a result, it is viewed as if it has been enlarged from the fluid motion, at the point where they meet with the matching grid motion algorithm, so as to evade deformed elements and also to minimize error due to discretization in the best way possible. According to the available literature, Arbitrary Lagrangian-Eulerian based, FSI approach can be dated back to a number of authors who pioneered this work such as Hirt et al, Belytschko et al among others (De Hart J, Peter J, S, and Belytschko, T, 2006). The most important advantage attributed to ALE based approach is: there exists a structural position in the fluid domain is referred to as a priori, which makes it possible to construct a perfect fine mesh close to structural surface and which is responsible of resolving the flow features around the prairie. Although in some cases, huge structure distortions can deform the fluid mesh, to an extent making mesh updating as well as re-meshing compulsory, and as a result making it impossible to preserve the optimal fluid mesh that surrounds the structure (Schott B, & Wall A, W,, 2014).

It is from these challenges that motivated the development of an alternative FSI approach, so as to try and address this challenge in the best way possible, the resulting approach is referred to as fixed grid FSI approaches or simply fixed grid methods. In this approach, as its name states, it uses static interfaces in tackling fluid problems, but it uses an Eulerian grid formulation in explaining fluid field, although it applies the same idea as in other fluid problems. This explains why it is possible to have complex and unlimited structure deformations ( Gamnitzer, P & Gerstenberger, A, 2008). The point at which the fluid contacts the structure or in other words, at the fluid structure interface, is normally explained fully or partially by the structure’s surface, for instance through a level set functions. In this case the fluid domain is divided into two; that is actual or real physical sub-domain, and the other is factitious or void fluid sub-domain, which is under structural field and does not have any meaning (Gamnitzer, P., Gerstenberger, A, and Forster, C., 2006). Depending on the actual fixed grid approach, there are a number of ways in which the void fluid sub-domain may be treated, for instant, the coupling conditions at the point where the fluid contacts the structure requires to be weakly enforced, due to misalignment that fluid-structure has with the element boundaries, although it is located within the interiors of the fluid elements.

There are two classes of fixed grid FSI approaches, and these are: one, we have, immersed boundary method which was developed purposely for muscle contraction and blood flow simulation by Peskin, the other one is Distributed Lagrange Multiplier / Fictitious domain method which have some similarities with Immersed Boundaries and explained in details by De Hart et al, among others.

In the case of Distributed Lagrange Multiplier/ Fictitious Domain, abbreviated as DLM/FD, the approach has a volumetric coupling which takes place between the structure and the fluid in other words the coupling of the fluid-structure takes place between the structure and the void fluid sub-domain (Gee, M and Kuettler, U., 2011). Monolithic system, therefore, is used to solve the structure’s degree of freedom as well as those of the fluid, and the kinematic coupling that exists between the structure and the void fluid sub-domain is enforced by the surplus Lagrange multiplier (Kuettler U, Gee M, W, and Wall A, W., 2011). The two approaches are also termed as unappropriated for solving FSI problems since they dot give accurate results as it is required especially when determining interface quantities. In addition to this Immersed boundary is dependent on mesh size which exists between the structure and the fluid and in which its absence can hinder achieving a perfect kinematic matching. As it is stated by Wall et al, another effect is that since the structure and the fluid interaction has what is called volumetric coupling, the artificial viscosity is affected by the structure or made to be incompressible.

Wall and Gerstenberger were moved by these challenges and as a result, they developed a fixed grid approach that utilized two dimensional FSI, which was based on the Extended Finite Element method abbreviated as XFEM, and along the interface the coupling is enforced weakly by the use of the Lagrange Multipliers. This type of FSI approach (XFEM) has the ability to take the position of a sharp at the point that links the fictitious fluid and physical sub-domains and as a result, such discontinuities such as pressure fields and velocity as well as derivatives such as stress all over the fluid structure interface are accurately represented (Gerstenberger A, and Wall A, W, 2012).

The fluid to the fluid system in conjunction with structural equations requires a solution which will be in the form of the monolithic scheme in other words; it does not require iterations at the middle of the fixed grid background liquid, the structure and fluid patch. Although, when the moving fluid patch as well as Eulerian background fluid are combined they become even more complex and as a result this nonlinear problem can be solved monolithically by the help of Newton-Raphson scheme ( Mok, D, & Wall, A, W., 2012). This takes us to a global system which alters the systems degree of freedom in each Newton step, since, in each and every new location of fluid to fluid interface, there arises a new fictitious and physical background fluid mesh and as a result, some degrees of freedom are lost and at the same time some are developed in each Newton step (Forster, C., 2007). In addition, the values that were lost in the previous or former Newton steps, and the present newton step requires it, they are re-constructed through an approach referred to as XFEM time integration. This process eliminates convergence, and it is also known to reduce the chances of having a poor convergence when using Newton-Raphson scheme and as a result an algorithm is developed so as to deal with this challenge. According to Shahmiri, monolithic schemes take the whole fluid structure into consideration and as it is explained above it solves for each and every degree of freedom at once, on the other hand, the iterative method, gives room for tackling the structure and the fluid fields differently and force an equilibrium state by exchanging iteratively the surface displacements (Shahmiri, S and Schott B., 2014).

2.6 Immersed domain method

Immersed domain method is more accurate as it is compared to Immersed boundary method and the main objective for introducing immersed domain method was to solve these inaccuracy problems (Taira & Colonius, 2007). In a nutshell, the immersed boundary method works with structures such as 2D space in a closed curve or even fires and a 3 dimensional space since they do not occupy volume, although Peskin records that for the bodies that occupy volume when immersed, a network of linked fibers can be used during approximation, and in this case each fiber is assumed to be an immersed boundary (Peskin C, 2011). This is the main cause of inaccuracy which poses as a demerit in this approach, where accurate modelling of the fluid motion doesn’t achieve a realistic structural response. There are advantages that come with an immersed domain method which covers structural domain with an introduced artificial fluid and as a result the fluid domain is enlarged so as to involve the whole computational domain. Due to the presence of the artificial fluid domain, there exists a condition which inhabits slipping and as a result the velocity and position between the local fluid and the immersed structure are maintained ( Forster, C & Wall, A, W., 2007). For this grip to be maintained, an FSI force is applied to each grid location within the artificial fluid domain and also at the point where the fluid meets the structure commonly known as fluid structure interface. For one to get the velocity equation of the whole domain, he would be required to solve the fluid equation, and from this, the velocity as well as the structural displacement will be known, and therefore one can obtain FSI force by substituting the values on the structural constitutive law, which is later utilized in determining the new velocity.

The determined fluid velocity is extended so as to capture structural domain, whose condition is based on non-slip or fixed, then using the structural velocity is used to update the structural configuration. Chang, Lee, and Choi et al, improved the accuracy of the fluid structure interface by replacing the delta functions by (DCELM) when they were simulating motion of a rigid body where they applied the immersed finite body method. According to Chang and Lee, the most essential assumption in this method of immersing domain is the fact that the structure is nearly incompressible or is purely incompressible, this follows the fact that the immersed structure has to keep or maintain the similar velocity constraint as it is with the incompressible fluid that surrounds it (Wang C, Xho L, and Xu C., 2006). In many problems that involve FSI approach, the structure’s volumetric strain may be too low or the structure’s volume may be significantly smaller as it is compared to the volume of the fluid and as a result this issue of the incompressibility condition can partially be solved. Although in situations where we have acoustic FSI challenges this assumption may not work, in other words, we are referring to a situation where both the structure and the fluid needs to be modelled so as to be compressible materials (Gee, M, 2010.). The main aim of having immersed domain method is to solve geometrically complex domains which pose the primary challenge in a number of modern mechanics computation applications. Immersed domain analysis, quests to achieve an accurate use of moving or evolving domains in a fluid mechanic computation technique.

2.7 Other immersed methods

Researchers have continued to examine the related numerical techniques that differ from the immersed boundary methods (Astirino, M, Franz , C & Miguel, A, 2015). The leakage problems are associated with the original immersed boundary method. The only option is that mass conservation can be improved by the use of free-finite-difference operators (Rossi, R & Onate, E., 2015). It is worth noting that the immersed interface method was developed by early researchers with the aim of improving the volume of conservation (Richter, T & Wick, T, 2015). Just like the immersed boundary method, the FSI force is computed from the structural configuration in the immersed interface method such that the boundary force is then used to jump the conditions in the pressure and the normal derivatives in the velocity (Takizawa, K, Yiri, B. & Tayfun, E., 2015). The immersed interface method incorporates the jumps in solution to produce second order approximations in finite difference schemes as opposed to the use of discrete delta functions seen in the immersed interface methods (Ceniceros, Fisher & Roma, 2009). Just like the immersed interface method that is limited to structures without volumes, the derivation of the jump conditions requires that the immersed structures be a 3D closed surface or a 2D closed curve (Nobile, F & Vergara, C, 2008). Researchers have continued to look into various methods of immersing interface methods with mathematical analysis and the examination of its accuracy (Liu, Y, Takeo, T & Maricus, T,, 2013). In order to stimulate the fluid motion within the immersed structures, the direct forcing method was proposed in the past decade (Lie, S, T and Yu G,, 2002). Lie and Yu noticed that this method directly evaluates the FSI force from the fluid equations by incorporation the known structural interfacial velocity by simply imposing the non-slip condition in the process. This method utilizes the computational force with non-zero values near the interface in solving the fluid equations in the entire fluid domain (De Tullion et al, 2007). Lie and Yi noticed also noticed that this method is beneficial in that it avoids the numerical stiffness encountered in penalty forcing techniques and this improves its accuracy. Gu and Wang also observed that the direct forcing method can be can also be implemented in an implicit manner such that the fluidic and the structural velocities are passed through a large coupled system simultaneously. They proposed two numerical methods to overcome the difficulty of locally preserving the mass of flow.

One of the methods suggested by Kevlahan and Oleg is the mirroring immersed boundary method that takes maximizes on the numerical stability and the efficiency of the entire system (Stockie & Green, 2006). In addition, it has been noticed that the mirror immersed boundary method identifies an interior point together with an exterior point near the immersed boundary that is assumed to be a closed surface that is used to determine the velocity of the interior point (Lassila, T, Alfio, Q, & Gianluigi, R., 2012). The exterior paint provides the known velocities that are substituted to determine the exterior fluid field in the fluid momentum equations. The other method suggested by Ervedoza and Vanninathan is the distributed Lagrange multiplier method that is further classified into different groups, depending on the constraint conditions incorporated into the solution and the time of FSI equations. The results of the FSI governing equations are subject to both time and space to release a set of algebraic equations of velocity, pressure. These can then be subjected into field equations to form an augmented matrix equation with several unknown Lagrange multipliers. Most researchers have utilized this method in solving the FSI problems.

The other group of the distributed Lagrange multiplier approaches the management involve the discretization of the field equations in space to form a set of ordinary equations of velocity that are then subjected to velocity constraints. According to Gerbeau and Marina, this situation employs the use of fractional time-stepping methods of gradually adjusting the field solution that satisfies the constraints such that most of these methods are first order accurate (Gerbeau, J and Marina, V., 2003).

The research carried out in the past decade helps to solve the problem with many rigid bodies that move in an incompressible flow (Deparis, S, Miguel, A, & Luca, F., 2003). Such a problem is avoided by the repelling the force between the rigid bodies due from collusion by adding the equation describing the rigid motion. Initially, such equations are represented in weak form that represents their fluid and structural domains. This method extends the fluid domains to such that it provides the cover to the rigid domains to make the fluid velocity at the same rate with the rigid body velocity, enforced by the means of Lagrange multipliers (Degroote, J, 2013). The spanning of the entire computational domain is made possible by the extension of the artificial fluid domain and the motion of the artificial fluid is accounted by the rigid body equally of the motion.

The sub-iterations that improve the numerical stability of an explicit time-stepping method were also proposed by Boulakia, m and Axel, O, (2008). This sub-iteration formulation extends the fluid domain to the cover domain of the particle by imposing the Dirichlet interface condition on the fluid particle interface. In the Dirichlet interface, the particle is initially modelled as an elastic body before introducing the rigidity of the constraint with the aim of forcing the strain rate in the particle to be zero (Crosetto, P, et al, 2011). However, the weak form of the system excludes the Dirichlet interface condition that is only imposed on the fluid particle interface. The source item is then added to the fluid momentum equation of the entire domain in the numerical procedure. With relation to the Lagrange multipliers associated with the rigidity constraint, the source item is the divergence of the strain rate. Moreover, the computed fluid velocity is the first projected to a divergence-free field before the rigid body motion of particles as per the fractional step method. According to Boulakia and Axel, the difference between the divergence-free velocity yields and the rigid body velocity are related to the source of the materials (Boulakia, m and Axel, O, 2008).

Badia, Annalisa, and Alfio applied the distributed Lagrange multiplier method in solving the motion of a non-linear elastic body immersed in a fluid of the domain (Badia, Annalisa, and Alfio, 2008). In this regard, the elastic body for small strains is approximately incompressible when the incompressibility condition is imposed onto the fictitious domain without affecting the structural displacement. The Lagrange multiplier in each finite element is then interpolated by the continuous bilinear shape functions (Astirino, M, Franz , C & Miguel, A, 2015). In order to maintain the numerical stability of the operation, the associated integration over the element is done with the trapezoidal rule since the resultant method is often unstable. The integration method forms a scheme that is used to form a stable, accurate and first order method before solving the structure of the equation to give the boundary location.

2.8 Moving implicit fluid surfaces

Boulakia and Axel argue that the position of the domain boundary depends on *x* and *t* in case of transient FSI problems. Moreover, Degroote also observes that the problem associated with the fixed grid methods with moving implicit interfaces can be linked with the FE space-time formulation or an Eulerian fluid formation with traditional finite difference time-stepping. Some scientists have proposed a space time method for implicit FSI interfaces. According to Kevlahan and Oleg, the benefit of space-time formulation is that it provides a consistent formulation of the evolving implicit interface in the space and time as it takes the advantage of the domain integration on the cells (Kevlahan, N, and Oleg, V, 2005). The important characteristic of the XFEM domain integration cells that researchers take advantage of is that it is aligned with the interface of space and time and helps to avoid ambiguities that could result from the stepping techniques (Lassila, T, Alfio, Q, & Gianluigi, R., 2012). However, researchers argue that space and time require additional computational effort and enhanced computer memory that makes the process challenging. The ALE formulation allows a reference movement that is non-reliable to the convective fluid formation for finite-difference time-stepping. Kevlan and Ole also observed that the benefit of the reference movement is that it allows the fluid mesh to follows the structure on the surface with much ease. They argue that the strong deformation of the reference mash requires re-meshing and this instigates the primary motivation for the research in the fixed grid methods. Researchers have observed the implicit interfaces at the time step that intersect the fluids elements and the integration of the cells that are being generated accordingly while taking the advantage that the fluid elements and the integration cells follow the interface in the ALE formulation at certain distances. Rossi and Onate observed that ALE approach reduces to the Eulerian formulation when the mesh velocity is zero since the velocities and accelerations are projected from the deformed fluid mesh onto the deformed background. The velocities and accelerations are projected from the deformed fluid mesh onto the non-deformed background grid to allow the intersection of the layer of the fluid elements. The project velocity is discretely incompressible for the new interface configuration if the projection is combined with additional constraints. Richter and Wick also noted that the projection between the non-fitting meshes are now required at every moment of step near the interface despite it affecting a relatively small domain along the implicit interface. In this regard, the errors introduced by the system must be considered for the overall accuracy and reliability of the transient solution.

It has also been observed that the Eulerian description of the flow field has a long tradition of FV methods as influenced by the works of traditional researchers in this field such as Fedkiw and Sussman. The main observation of such experiments is that it is affected by time discretization and this affects the conservative equations presented in the diagram (Lie, S, T and Yu G,, 2002). From the experimental point of view, it is questionable to transform the integral equations of momentum conservation and mass directly into their differential equations. These transformed differential equations are needed as the beginning point for the weighted residual based finite formulation (Stockie and Wetton, 2009). Gu and Wang also proposed the Ghost fluid method to help sustain the specific needs of the fixed grid FSI approach with various boundaries. These velocity conditions can be applied weakly along the interfaces as part of the moving interface. Several assumptions need to be made in this arrangement since the smooth movement of the surface of the observed domain is assumed in time (Gu, Y, T and Wang, Q, 2008). In this regard, it allows for the reasonable estimates for the old time step values for the newly uncovered nodes.

2.9 Computational Fluid-Structure Interactions method

In the recent years, much attention has been given to fluid-structure interaction problems and different multi-field problems. It is believed that their importance is ever growing because they considered being of great relevance in the field of engineering (Wall A, W., 2010). As of late, research in the zone of computational fluid structure interaction has greatly progressed. Computational liquid flow has turned into a crucial device for a great number of specialists. Computational liquid flow reproductions give understanding into the points of interest of how items and procedures function, and permit new items to be assessed in the PC, even before models have been manufactured. It is likewise effectively utilized for issue shooting and advancement. The turnover time for a computational liquid flow investigation is consistently being decreased since PCs are turning out to be constantly effective and programming uses continually proficient calculations. Ease, acceptable exactness and short lead times permit computational liquid flow to contend with building physical models, i.e. ‘virtual prototyping’. There are numerous business programs accessible, which have turned out to be anything but difficult to utilize, and with numerous default settings, so that even an unpractised client can get dependable results for basic issues. Be that as it may, most applications oblige a more profound comprehension of liquid progress, numeric and displaying. Since no models are all inclusive, computational liquid flow designers need to figure out which models are most proper to the specific case. Besides, this more profound learning is needed since it gives the talented designer the ability to judge the potential absence of precision in a computational liquid flow investigation. This is essential since the investigation results are regularly used to settle on choices about what models and procedures to assemble.

One point of interest utilizing computational liquid flow is that it is conceivable to acquire definite neighbourhood data in the recreated framework. In a fluidized bed it is conceivable to reproduce the change as well as the nearby temperature, the entrainment of particles, the back mixing and air pocket arrangement. This nitty gritty data will help with building a subjective comprehension of the procedure, and a parameter study can uncover extra data, for example, the jug necks and the operational furthest reaches of the gear.

The simulation of computational liquid flow without fitting information can be an exceptionally unverifiable device. The business computational liquid flow projects have numerous default settings and will quite often give results from the reproductions, be that as it may, to acquire solid results, the model must be picked with a legitimate procedure. A met arrangement shows the aftereffects of the particular picked model with the given lattice; in any case, it doesn’t uncover the truth. Without fitting comprehension of the computational liquid flow system and the demonstrating hypothesis behind it, computational liquid flow can get to be restricted to ‘vivid liquid showcase’. To comprehend the simulation procedure and the strides included in it, consider a sample of a course through a funnel twist. For a fluid to flow through a channel twist there is geometry developed, isolated into little parts/sections, called a lattice. With this work one can really characterize the test focuses where he needs the analysis to be finished. At that point, characterize the limit conditions to get a special arrangement, explaining it with a PC. The outcomes acquired gives a considerable measure of information along these test focuses that are then post-handled with representation devices to break down the outcomes.

There are numerous computational liquid flow programs’ commercial general-purpose, for instance CFX, Star-CD, Phoenics and Fluent. There are additionally some extremely concentrated projects mimicking burning in motors, cooling of semiconductors and reproduction of climate. An exceptionally valuable open-source program that can deal with most computational liquid flow issues is Open Foam. In any case, the documentation and the client interface do not also grown as those for the commercial codes. Commercial computational liquid flow bundles contain modules for CAD drawing, cross section, stream re-enactments and post-preparing. In tackling an issue utilizing computational liquid flow there are numerous strides that must be characterized.

3.0 *Least-squares methods for solving FSI problems*

The Least-Squares Finite Element Method has gotten broad thought as of late. The strategy is taking into account minimizing the *L*_{2} standard of the residuals delivered from the Least-Squares Finite Element Method rough guess of differential equations’ systems ( Kumar D, & Yeonseung R,, 2012). The Galerkin methodology’s weak-form is usually utilized as a part of standard finite element details. Unfortunately, the Galerkin methodology presents challenges when connected to non-self-adjoint comparisons on issues, for example, fluid flow and other transport issues (Rasmussen, 2009). These challenges incorporate the solution’s instabilities and oscillations. This can also lead to derivatives being poorly approximated. The Least-Squares Finite Element Method has gotten a lot of consideration as of late as a result of its capability to keep away from these challenges.

A significant point of interest of Least-Squares Finite Element Method is that its plan dependably prompts a symmetric algebraic equations that has positive-definite system, notwithstanding for non-self-adjoint system. This offers incredible favourable circumstances from a computational perspective. The utilization of powerful iterative strategies to tackle the arrangement of comparisons grew through Least-Squares Finite Element Method gets to be conceivable. Likewise, iterative arrangement strategies, for example, preconditioned conjugate inclination systems can be executed without the need of worldwide get together. For this technique, extensive scale issues can be unravelled utilizing a completely parallel environment and without the need of worldwide get together.

Least-Squares Finite Element Method has likewise been demonstrated to give more noteworthy exactness to the subordinates of primal variables than conventional finite elements taking into account Weak Galerkin techniques. These subordinates, frequently alluded to as auxiliary variables, are the reaction that is most regularly shared between areas for FSI issues. This gives a particular motivation to utilize Least-Squares Finite Element Method in coupled issues. The precision of the auxiliary variables at all squares plans starting from the execution of blended systems for Least-Squares Finite Element Method. Blended techniques use both primal and auxiliary reactions at the immediate level of opportunity reactions. This increases the aggregate number of framework degrees of flexibility, however the upgraded exactness and capacity to straightforwardly share and amass the optional degrees of opportunity for several problems enhances the precision at the interface which ought to enhance the accuracy of the arrangement general. The finite element method has generally been the numerical arrangement procedure of decision for basic issues. Utilization of the finite element strategy in different issues, for example, fluids, is a subject of incredible fixation as of late. Customary Weak Galerkin finite element method has demonstrated trouble tackling some non-auxiliary issues. Since Least-Squares Finite Element Method handles fluid and transport issues with fewer issues in some cases, the finite element strategy, taking into account slightest squares, may be utilized as a numerical close estimation procedure for an extensive variety of problems.

The Weak Galerkin finite element method has long been viewed as the conventional finite element system for decision due to its high arrangement precision and its low differentiability prerequisites on its shape capacities. A Strong Galerkin methodology is not as ordinarily utilized in light of the fact that it obliges full differentiability of the shape capacities. For illustration, a fourth-arrange the differential equation would oblige shape works that are fourth-arrange differentiable for a Strong Galerkin approach while a Weak Galerkin methodology would just oblige that the shape capacities are second-arrange differentiable. This decrease in differentiability prerequisites is on the grounds that the Weak Galerkin approach applies incorporation by parts to the first Galerkin useful.

Least-squares finite-elements were inspected strongly in the 1970s. For a period after that, slightest squares finite components were not a usually utilized plan. The technique did not get more thought until late years. This was for the most part in view of the acknowledgment that the higher polynomial request of the finite element shape capacities is a fundamental piece of utilizing Least-Squares Finite Element Method. The utilization of higher-request p-components determined the fundamental concerns brought up in the 1970s. After this, the strategy of Least-Squares Finite Element Method did not get more thought until late years. This was basically as a result of the acknowledgment that the higher polynomial request of the finite component shape capacities is a key piece of utilizing Least-Squares Finite Element Method.

3.1 Summary of the Literature review

Inasmuch as several methods for solving FSI problems have been investigated and presented in the literature review, further objectives and aims laid in this thesis requires more exploration. Just to mention a few, the research questions adopted in this paper are:

- Combining the benefits from the ALE grid approach and the fixed grid approach, by developing an overlapping grid formulation, will the localised ALE grid conforms to the structural deformation?
- Will the continuity and interaction between the fixed fluid grid and ALE fluid grid need to be enforced in the overlapping grid formulation?

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