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Partitioned approach for Fluid-Structure-Interaction (FSI) Atanas Gegov, TUM < [email protected] > Ferienakademie- Autumn 2007 What is FSI Different approaches for solving FSI problems Algorithmical improvements of the partitioned approach How partitioned FSI can be realized – FSI*ce Bibliography What is FSI FSI (German: “Fluid- Struktur- Wechselwirkung”) describes the phenomenon of interaction between fluid (liquid or gas) and solid body (structure) in a system. a) Why is FSI simulation interesting To determine the effects of fluid-structure interaction for a given system, engineering design often involves extensive experimental testing. However, experiments may be costly, time consuming and in some cases even infeasible. As numerical models and techniques have matured over the last decades to deliver more accurate predictions, and with the advent of increasing computing power for affordable prices, numerical simulation has become more established in the design process to support or even replace experimental testing. b) Examples of different FSI problems One of the most prominent when talking about FSI. The collapse of the Tacoma Narrows Bridge in 1940. The hydraulic ram pump. Pumping is possible without maintenance and power supply! Flow around elastic structures (airplanes, submarines etc.). Examples for three methods for describing fluids: Lagrangian, Eulerian, ALE (Arbitrary Lagrangian- Eulerian) description: Different approaches for solving FSI problems a) Monolithic approach Monolithic solution methods treat the coupled fluid and structure equations simultaneously, i.e., they directly operate on the aggregated fluid and structure equations. As this system is in general nonlinear, the solution procedure typically involves a Newton method. b) Partitioned approach Very popular in solving FSI. It is not only used in FSI but the idea can be applied to different problems and situations. Important applications are provided by thermomechanics, fluid-structure interaction and control-structure interaction. Idea: In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately 1 advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. The partitions interact with each other on their interface (typically a mesh structure that is closed, for example airplane). The term partitioning identifies the process of spatial separation of a discrete model into interacting components generically called partitions. The decomposition may be driven by physical, functional, or computational considerations. For example, the structure of a complete airplane can be decomposed into substructures such as wings and fuselage according to function. Substructures can be further decomposed into submeshes or subdomains to accomodate parallel computing requirements. Subdomains are composed of individual elements. Going the other way, if that flexible airplane is part of a flight simulation, a top-level partition driven by physics is into fluid and structure models. This kind of multilevel partition hierarchy: coupled system, structure, substructure, subdomain and element, is typical of present practice in modeling and computational technology. Terminology: A coupled system is one in which physically or computationally heterogeneous mechanical components interact dynamically. Physical Subsystems. Subsystems are called physical fields when their mathematical model is described by field equations. Examples are mechanical and non-mechanical objects treated by continuum theories: solids, fluids, heat, electromagnetics. Artificial Subsystems. Sometimes artificial subsystems are incorporated for computational convenience. A coupled system is characterized as two-field, three-field, etc., according to the number of different fields that appear in the first-level decomposition. For computational treatment of a dynamical coupled system, fields are discretized in space and time. A field partition is a field-by-field decomposition of the space discretization. A splitting is a decomposition of the time discretization of a field within its time step interval. Pros and contras: Customization. This means that each field can be treated by discretization techniques and solution algorithms that are known to perform well for the isolated system. Independent Modeling. The partitioned approach facilitates the use of non-matching models. Software Reuse. The partitioned approach facilitates taking advantage of existing code. Modularity. New methods and models may be used in a modular fashion according to project needs. Disadvantages: The partitioned approach requires careful formulation and implementation to avoid serious degradation in stability and accuracy. Parallel implementations are particularly delicate. In summary, circumstances that favor the partitioned approach for tackling a new coupled problem are: a research environment with few delivery constraints, access to existing software, localized interaction effects (e.g. surface versus volume), and widespread spatial/temporal component characteristics. The opposite circumstances favor a monolithic approach. Loosely- coupled and strongly- coupled partitioned approaches: If only a single (one time for the fluid program and one for the structure) solution per time step are carried out, such partitioned methods are commonly referred to as loosely-coupled partitioned methods. Their essential disadvantage pertains to the loss of the conservation properties of the continuum fluid-structure system. Although the order of the incurred error can be improved by predictors, loosely-coupled methods can never be exactly conservative. Partitioned methods which solve the fluid-structure system by repeating within a time step alternate fluid and structure solutions until convergence are called strongly-coupled partitioned methods. 2 To improve the order of the numerical evaluation error incurred by loosely-coupled partitioned methods, prediction techniques are used. For example, instead of integrating the fluid equations based on the position of the structure boundary in the previous time interval, a prediction can be used for the position of the structure boundary in the current time interval. Strongly-coupled methods have a greater computational cost per time step than loosely-coupled methods. However, strongly-coupled methods can maintain conservation at the fluid-structure interface, which renders them unconditionally stable. In contrast, loosely-coupled methods are typically energy increasing and, hence, numerically unstable. Algorithmical improvements of the partitioned approach As mentioned above, the numerical solution of fluid-structure interaction problems commonly employs subiteration, i.e., fluid and structure equations are solved alternately according to the interface conditions. This process is repeated until convergence. Subiteration sidesteps a simultaneous treatment of the coupled fluidstructure equations and the resulting difficulties. The following steps define the subiteration process: for j = 1, 2 . . . (1) Solve the kinematic condition: Constitutes a boundary condition for the initialboundary-value problem of the fluid (2) Solve the fluid: Result is the flow velocity and pressure fields (3) Solve the dynamic condition: Result is the fluid pressure (the forces) acting on the structure surface (4) Solve the structure: Result is the displacement of every point on the structure surface. The subiteration process can then be seen as a mapping from one structural interface displacement to the next, i.e. C: zj → zj+1 = Czj. The fixed point is where .z: C.z = .z. Although subiteration is a good solver for many problems, it suffers from two essential drawbacks: Firstly, subiteration converges only slowly or even diverges for problems with large computational time steps. Subiteration is only conditionally stable. Secondly, subiteration is generally employed in a sequential time-integration process and, hence, solves a sequence of similar problems. However, the method cannot exploit this property and reuse generated information, for instance, for preconditioning purposes. Therefore, subiteration is to be considered inefficient. It is possible to overcome these drawbacks by combining subiteration with GMRES acceleration. A Multi-Grid solution method is another approach for achieving better efficiency in the subiterations. a) Multi-Grid It also makes subiterations, but the point here is that the iterations are done one more than one grids. One begins from the top-level (the main grid where the FSI has to be solved) and goes down to levels with lower resolution, where the iteration is accordingly less expensive due to the reduced dimension. Than the gathered information is propagated again to the top levels and makes therefore their iterations also more efficient. Although the idea of Multi-Grid is good, it is not so easy to be realized in practical applications. b) Interface-GMRES(R)/ Newton-Krylov The nonlinear problem Cz =z can be formulated as Cz-z=0 and Rz=0 with R=C-I. After some transformations we get R’ (z (i))*(z (i)-z (i+1)) =R (z (i)). One can see the shape A*x=b which is an LSE. This system can be solved be the GMRES method. GMRES approximates the exact solution of Ax = b by the vector xn e Kn that minimizes the norm of the residual Axn − b. The nth iterate minimizes the residual in the Krylov subspace Kn. The idea is that after a small number of iterations, the vector xn is already a good approximation to the exact solution. 3 With Interface- GMRES we are performing better than with the fix-point iteration, but there is still one more optimization possible. As each Newton step invokes the solution of a linear system by a Krylov method (GMRES), the Newton-Krylov method lends itself naturally to reuse of Krylov vectors in subsequent Newton steps. In our context we refer to such reuse as Interface-GMRESR. Typically, much fewer Krylov vectors need to be added to the reused space than are generated for a reconstructed Krylov space, which can result in considerable computational savings. The search in direction of the zero-point is not more “random” but more specific and directed. Therefore there are less subiterations needed and this means a significant increase in efficiency. It appears that the reuse of the Krylov vectors can make the computational expense of the Interface-GMRESR method comparable to loosely-coupled partitioned methods which perform only a single fluid and structure solution per time step. However, in contrast to such loosely-coupled methods, the Interface-GMRESR method enables conservation and improves stability and accuracy. How partitioned FSI can be realized – FSI*ce A real environment for making the partitioned FSI simulation on the computer. It is called FSI*ce and is developed by the teams of Prof. Rank and Prof. Bungartz (TUM). Requirements: This is firstly a “Plug-In” mechanism for the CFD/CSD programs (this means simple replacement ability for the components) and secondly- implementation of the coupling schema (for example Interface-GMRES or simply single iteration per time step) outside from the CFD/CSD simulation programs. Design: The full flexibility of the partitioned approach is achieved in FSIce with the Client-ServerModel. In FSIce a vertex-edge-face Graph (vef-Graph) is used for the coupling geometry. FSIce allows communication via MPI or Sockets. FSIce is already successfully tested with programs developed in scientific environment that allow access to the source code. However, FSIce is not completely developed and is going to be further improved. Bibliography Books: “Efficient Numerical Methods for Fluid-Structure Interaction” by Christian Michler, Netherlands 2005 Papers: “Partitioned analysis of coupled mechanical systems” by Carlos A. Felippa, K.C. Park, Charbel Farhat, USA 1999 Paper about FSIce (title to be defined) by TUM Lehrstuhl V (Dipl.-Geophys. Markus Brenk), Germany, to appear Internet: FSI in general: http://www.win.tue.nl/fsi/ Eulerian and Lagrangian fluid description: http://numerik.iwr.uni-heidelberg.de/Research/dunne.html Tacoma Narrows Bridge: http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge Hydraulic ram pump: http://schou.dk/animation/ Newton’s method: http://en.wikipedia.org/wiki/Newton's_method Partition solution of coupled systems: http://www.inf.bauwesen.tu-muenchen.de/~kollmannsberger/ SoftLab2005CoupledSystems/Files/third_presentation.ppt GMRES approach: http://de.wikipedia.org/wiki/GMRES-Verfahren or http://en.wikipedia.org/wiki/GMRES Krylov subspace: http://de.wikipedia.org/wiki/Krylow-Unterraum Linear span: http://de.wikipedia.org/wiki/Lineare_H%C3%BClle Forschergruppe 493: http://fsw.informatik.tu-muenchen.de/index.php MPI exercises: http://www-unix.mcs.anl.gov/mpi/tutorial/mpiexmpl/contents.html 4