Fluid-Structure Interaction - AbsoluteAstronomy.com

Fluid-structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow. Fluid-structure interactions can be stable or oscillatory. In oscillatory interactions, the strain induced in the solid structure causes it to move such that the source of strain is reduced, and the structure returns to its former state only for the process to repeat.

Examples

Fluid-structure interactions are a crucial consideration in the design of many engineering systems, e.g. aircraft and bridges. Failing to consider the effects of oscillatory interactions can be catastrophic, especially in structures comprising materials susceptible to fatigue

Fatigue (material)

'In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values are less than the ultimate tensile stress limit, and may be below the yield stress limit of the material.Fatigue occurs...

. Tacoma Narrows Bridge (1940)

Tacoma Narrows Bridge (1940)

The 1940 Tacoma Narrows Bridge was the first incarnation of the Tacoma Narrows Bridge, a suspension bridge in the U.S. state of Washington that spanned the Tacoma Narrows strait of Puget Sound between Tacoma and the Kitsap Peninsula. It opened to traffic on July 1, 1940, and dramatically collapsed...

, the first Tacoma Narrows Bridge, is probably one of the most infamous examples of large-scale failure. Aircraft wings and turbine blades can break due to FSI oscillations. Fluid-structure interaction has to be taken into account for the analysis of aneurysm

Aneurysm

An aneurysm or aneurism is a localized, blood-filled balloon-like bulge in the wall of a blood vessel. Aneurysms can commonly occur in arteries at the base of the brain and an aortic aneurysm occurs in the main artery carrying blood from the left ventricle of the heart...

s in large arteries and artificial heart valve

Artificial heart valve

An artificial heart valve is a device implanted in the heart of a patient with heart valvular disease. When one of the four heart valves malfunctions, the medical choice may be to replace the natural valve with an artificial valve. This requires open-heart surgery.Valves are integral to the normal...

s. A reed

Reed (instrument)

A reed is a thin strip of material which vibrates to produce a sound on a musical instrument. The reeds of most Woodwind instruments are made from Arundo donax or synthetic material; tuned reeds are made of metal or synthetics.-Single reeds:Single reeds are used on the mouthpieces of clarinets...

actually produces sound because the system of equations governing its dynamics has oscillatory solutions. The dynamic of reed valve

Reed valve

Reed valves are a type of check valve which restrict the flow of fluids to a single direction, opening and closing under changing pressure on each face...

s used in two strokes engines and compressors is governed by FSI. The act of "blowing a raspberry

Blowing a raspberry

Blowing a raspberry or strawberry or making a Bronx cheer is to make a noise signifying derision, real or feigned. It is made by placing the tongue between the lips and blowing, making a sound redolent of flatulence. In the terminology of phonetics, this sound can be described as an unvoiced...

" is another such example.

Analysis

Fluid-structure interaction problems and multiphysics

Multiphysics

Multiphysics treats simulations that involve multiple physical models or multiple simultaneous physical phenomena. For example, combining chemical kinetics and fluid mechanics or combining finite elements with molecular dynamics...

problems in general are often too complex to solve analytically and so they have to be analyzed by means of experiment

Experiment

An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

s or numerical simulation. Research in the fields of computational fluid dynamics

Computational fluid dynamics

Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

and computational structural dynamics

Structural Dynamics

Structural dynamics is a subset of structural analysis which covers the behaviour of structures subjected to dynamic loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structure can be subject to dynamic loading. Dynamic analysis can be used to find dynamic...

is still ongoing but the maturity of these fields enables numerical simulation of fluid-structure interaction. Two main approaches exist for the simulation of fluid-structure interaction problems:

Monolithic approach: the equations governing the flow and the displacement of the structure are solved simultaneously, with a single solver
Partitioned approach: the equations governing the flow and the displacement of the structure are solved separately, with two distinct solvers

The monolithic approach requires a code developed for this particular combination of physical problems whereas the partitioned approach preserves software modularity because an existing flow solver and structural solver are coupled. Moreover, the partitioned approach facilitates solution of the flow equations and the structural equations with different, possibly more efficient techniques which have been developed specifically for either flow equations or structural equations. On the other hand, development of stable and accurate coupling algorithm is required in partitioned simulations.

Numerical simulation

Both the Newton-Raphson method

Newton's method

In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...

and fixed point iteration can be used to solve FSI problems. Methods based on Newton-Raphson iteration are used in both the monolithic
and the partitioned approach. These methods solve the nonlinear flow equations and the structural equations in the entire fluid and solid domain with the Newton-Raphson method. The system of linear equations within the Newton-Raphson iteration can be solved without knowledge of the Jacobian with a matrix-free iterative method

Iterative method

In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...

, using a finite difference

Finite difference

A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...

approximation of the Jacobian-vector product.

Whereas Newton-Raphson methods solve the flow and structural problem for the state in the entire fluid and solid domain, it is also possible to reformulate an FSI problem as a system with only the degrees-of-freedom in the interface’s position as unknowns. This domain decomposition condenses the error of the FSI problem into a subspace related to the interface. The FSI problem can hence be written as either a root finding problem or a fixed point problem, with the interface’s position as unknowns.

Interface Newton-Raphson methods solve this root-finding problem with Newton-Raphson iterations, e.g. with an approximation of the Jacobian from a linear reduced-physics model . The interface quasi-Newton method with approximation for the inverse of the Jacobian from a least-squares model couples a black-box flow solver and structural solver by means of the information that has been gathered during the coupling iterations. This technique is based on the interface block quasi-Newton technique with an approximation for the Jacobians from least-squares models which reformulates the FSI problem as a system of equations with both the interface’s position and the stress distribution on the interface as unknowns. This system is solved with block quasi-Newton iterations of the Gauss-Seidel type and the Jacobians of the flow solver and structural solver are approximated by means of least-squares models.

The fixed-point problem can be solved with fixed-point iterations, also called (block) Gauss-Seidel iterations , which means that the flow problem and structural problem are solved successively until the change is smaller than the convergence criterion. However, the iterations converge slowly if at all, especially when the interaction between the fluid and the structure is strong due to a high fluid/structure density ratio or the incompressibility of the fluid . The convergence of the fixed point iterations can be stabilized and accelerated by Aitken relaxation and steepest descent relaxation, which adapt the relaxation factor in each iteration based on the previous iterations .

If the interaction between the fluid and the structure is weak, only one fixed-point iteration is required within each time step. These so-called staggered or loosely coupled methods do not enforce the equilibrium on the fluid-structure interface within a time step but they are suitable for the simulation of aeroelasticity

Aeroelasticity

Aeroelasticity is the science which studies the interactions among inertial, elastic, and aerodynamic forces. It was defined by Arthur Collar in 1947 as "the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural...

with a heavy and rather stiff structure.
Several studies have analyzed the stability of partitioned algorithms for the simulation of fluid-structure interaction

.

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