Fluid-Structures Interaction – HPC Perspective

Fluid-Structures Interaction – HPC Perspective
Prof. Gopal Shevare

Contents Fluid-structure interaction (FSI) and classification of FSI in aerospace Fluid-structure interaction as an HPC application Topic in Fluid-Structure Interaction Methods of solving fluid-structure interaction Reduced order modeling Portioning and staggered computations Interfacing CFD and FEM solvers Summary 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Analysis & Multi-Disciplinary Analysis
Fluid Dynamics: Calculation of flow variables. The walls of the body are rigid. The flow could be unsteady Dynamics: Motion of a rigid body under known loads Elasticity : Deformations of non-rigid body under steady forces and moments Multi-Disciplinary Analysis Flight Dynamics : Motion of rigid body under fluid dynamic loads Structural dynamics: Motion of elastic bodies under given unsteady loads Aero elasticity : Motion of body under fluid dynamic. loads which are affected by body deflections and velocity 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Fluid-Structure Interaction (Aero-Elasticity)
Aero-servo-elasticity involves fluid dynamics + structural dynamics + controls It is more complex field than aero-elasticity 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Analysis (Fluid Dynamic Analysis)
19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Flight Dynamics = Fluid Dynamics + Rigid Body Dynamics
Requires 100s of (meshing and CFD runs) Rigid body dynamics does not require meshing or solution of PDEs 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Flight Dynamics = Fluid Dynamics + Rigid Body Dynamics
Will require 100s (CFD meshes and CFD runs) x 100s (FEM meshes and FEM runs) 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Classification of Fluid-Structure Interactions
Divergence : Catastrophic static deflections under fluid dynamic loads Flutter : Self feeding, through positive feedback, destructive vibration. If aerodynamic excitation is more than natural damping, vibration amplitude increases. Mild flutter is also called buzz Buffeting : When separated flow impinges in structure, the structure vibrates causing buffeting Limit Cycle Oscillations (LCO) : Typically happens when the flow itself is unsteady, may be with hysteresis Control reversal: Unintended / reverse response to the control due to large deflection of the structure 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Classical views on Flutter
Flutter requires synchronized interaction between two modes. One mode absorbs energy from the airflow, the other absorbs and gains the amplitude Typically, bending motion absorbs energy and torsion modes amplifies to store it The modes must have the same frequency so that only one combined mode is possible With large enough airspeed failure occurs – flutter speed Most two most importation causes are: Unfavorable ratio of wing torsion stiffness to wing bending stiffness, Presence of mass at point at unfavorable position 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Classical Methods of Flutter Analysis [1]

Motivation - Standard Validation in FSI

Motivation : Scatter in the Accuracy of Prediction

Computational Methods for Aeroelastcity

Governing Equations for FSI
The Governing equations for “Dynamic Fluid Structure Interaction (FSI or DFSI) ” are three field problems: The Navier–Stokes equation for compressible flow In ALE formulation (dynamic meshes), the velocity is expressed relative to the movement of the mesh, 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Governing Equations for FSI
The Governing equations for the motion of a structure is Damping within the structure, C is approximated by Rayleigh damping as a linear combination of the stiffness and mass matrices Mesh movement is modeled as a fictitious pseudo structural problem having its own dynamics and is governed by Km is pseudo stiffness matrix defined for the whole mesh; dm is the displacement of structural mesh 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Classification of Multi-Disciplinary Analysis Methods
Inter-field - a property of all partitions Intra-field – a property of only one partition 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Code Development Process
Multi-physics code would be too difficult / too slow 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Partition Approach [2] for Solution of FSI Problems

Staggered Partition Approach
This is heterogeneous & coupled methodology It consists of partitioning in space and splitting in time 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Comments on the Methods…
Monolithic methods (also called simultaneous treatment method) : The system is treated as a monolithic problem and all variables are advanced simultaneously in time. Partition treatment: The fields are computed separately and advanced separately, either simultaneously or in multiple time steps. Inter-fields belonging to the another partitions are considered as forcing functions No technical universal argument can be made for the overall superiority of either Their relative merits are not only problem dependent, but are inter-wined with human factors 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Comments on the Methods…
In many cases two fields can not exit in the same space. In such cases, separate fields may leads non-matched meshes. This is typical of fluid-structure interaction problems Separate PDEs for separate fields may have different stiffness, in which case, we need interfaces (often called frames) for transfer of information between subsystems in time. Thus the solvers are split in individual field time steps t field different from the given global time tglobal (or even pseudo time) Inter-field data happens across space and / or frames 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Advantage Partition Methods…
Each field can be treated by the best available algorithm. If the interaction effects are taken care of properly, the algorithm remains accurate / efficient Partition methods facilitate the use of non-matching models. Separate models can be built by different design teams / subcontractors / geographically separated groups The partitioned approach facilitates taking advantage of existing, private, public or commercial software. This is probably the best methods in fast changing computing technology Modularity: Since the codes are small, new developments can be introduced in the pool of tools according to needs keeping testing and validation incremental Flexibility and modularity is good for R&D Monolithic approach application development is possible if the individuals in the development team are domain experts in all the fields involved 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Computational Complexity Aeroelasticity
In MDA and MDO such as aero-elasticity, simulation run time is unacceptable In FSI, contribution to the no of degrees of freedom from CFD is more than 90%, from FEM (structures) less than 10 % Estimates of no of CFD simulations for flutter: 10 a/c configurations 50 flight points 100 mass cases 5 maneuvers 20 gusts (gradient lengths) 4 control laws ~20,000,000 CFD simulations Using engineering experience no of simulation can be reduced to ~100,000 Can we use some method which reduces simulation time drastically with some sacrifice in accuracy ? Source : AIAA 2008, Rossow, Kroll : Aero Data Production A380 Wing 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Time Staggered Partitioned approach for Complex Simulations such as Aeroelastcity 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Complex Systems Involve Four Multies
Several dynamic engineering phenomena involve system analysis requiring mastery over the following multies: Multi-scale: the space / time scales differ by an order of magnitudes - hydrodynamic vs. acoustic pressure Multi-physics: involves interaction of different behavior, as in structures and fluids Multi-processing: computational methods that use HPC systems where numerical algorithms are decomposed methods to achieve concurrency Multi-group: Activity needs more than one individual (or a group) for analysis and/or appreciation of the simulations being carried out Flutter is a complex system involving all the four multies 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

The Effect of Multies – an Example
Multies produce exponentially increasing options in simulations as each physics has evolved and matured with time on its own For example, common terminology acquires different meaning with fluid dynamics and structural dynamics 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Basic Guidelines for Complex Simulations
Define efficiency of simulations as follows:. Use partitioned closely coupled solvers Use time-staggerd partition approach through time splits as it is preferable for several reasons Use Reduced models (ROM) rather first-principle based partial differential equations for producing order of magnitude simulation efficiency in individual models Off-line work simulations are acceptable, even if additional, if they increases efficiency of simulation Fictitious partition for transfer of intra and inter data between fields 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Reduced Order Models for Fluid Dynamics and Structural Dynamics

Use of Reduced Order Models…
Consider CFD; it is a computationally demanding method in terms of data storage, data transfer and computing time The problem is very acute when multiple CFD simulations are required in multi- disciplinary simulations or multi-disciplinary design optimization Reduced order models overcome this difficulty ROM are ordinary differential equations, derived from experimental data, simulation data or the governing partial differential equations ROM contain fewer degrees of freedom than the discretized partial differential equations. They are therefore relatively inexpensive to compute. They are less accurate but cheaper compared to solving original system of PDEs 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Degrees of Freedom in CFD and FEM
CFD or FEM drive residue R to zero. R is given by their respective PDEs Both FVM and FEM use weighted residual (wi, RN) in  to solve the equations No of degree of freedom = mesh points (or no of cells) x no of variables 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Degrees of Freedom in ROM
Reduced order models (ROMs) approximate the solutions using global basis functions The global basis functions need to be optimal in the sense that “given their total numbers, they describe the field with the least error” Linear combination of chosen global basis functions gives the solution to steady or unsteady problem The no of degrees of freedom are an order of magnitude smaller and hence ROMs are extremely efficient In multi-disciplinary simulations, ROMs are extensively used 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Reduced Order Model – a 2D Example
99.9% of kinetic energy of the flow is can be captured by suing approximation: 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

ROM in CFD - How to get the Basis?
Global basis functions are extracted from an elegant and powerful data reduction technique called proper orthogonal decomposition (POD). Such global basis functions are called POD modes Properties of POD suggest that it is a preferred basis in many applications POD modes are optimal in capturing essence of infinite dimensional process in “few” modes Reduced order models with POD modes are reliable and optimal in providing detailed knowledge of the physics at reduced cost POD modes are useful in understanding the physics of the problem when underlying physics is too complex 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Deriving ROM in CFD Assume that flow-field is approximated as a linear combination of the basis functions Assume that j , j = 1, … M << N (no of mesh points x no of variables per mesh point) are known and are problem dependent. They can be obtained by carrying out high fidelity CFD simulations ROM are coupled ordinary differential equations for ai (t), i = 1,2,…, M in nos. The ODE for aj are found out by projecting the solution of full governing onto the reduced basis 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

General Formulation in ROM [5,6]
Consider a general PDE governing a physical phenomenon The operators L, N2 and N3 are linear, quadratic non-linear and cubic non-linear operators The Galerkin projection of this equation onto each POD mode φj is: Substitute the POD decomposition for u into the above equations and apply (a) the algebraic rules of inner products and (b) orthogonality to give ROMs as 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

General Formulation in ROM
These are system of ODE’s of order equal to no of retained POD modes The inner products in are functionals of the known, time-independent POD modes. These inner products can pre-computed before integration of the ROM In aero-elasticity applications of POD to build the reduced order model in terms of fluctuations over and above the respective steady state 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Model for Structural Dynamics
Semi-discrete Governing equations for structural dynamics are: M, C and K are the mass, damping and stiffness matrices, respectively, f = f(t) is the applied force and d = d(t) the state-vector, the response of the system This second order system can be reduced to first order through the auxiliary vector The resulting first order system is: A (non-singular) and B are called the weighting matrices. They are suitably chosen to reduce computations required 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Model for Structural Dynamics
The two choices for A and B are If A and B are Jensen and if C = 0, it can be seen that u is momentum, p If C  0, u becomes modified momentum vector which be interpreted as the dynamic force resisted by inertia and damping effects. Linear multi-step operator can be used to get the solution at tn+1, assuming that the solution is known at time stations t0,... tn, with n ≥ m − 1. 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Time Discretisation for Structural Dynamics
 and β depend on the chosen finite difference scheme If β coefficients associated with u are nonzero, the scheme implicit After some rearrangements, the coupled system of ODEs can be cast in canonical form: 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Finite Difference Equations for Staggered Partitioning Computations
It is not good enough to have a set of first order ODEs ROM for fluid dynamics or structural dynamics, the scalability of this multi-physics problem must be ensured on currant day HPCs The no of ODEs could be thousands. But note that without ROM, the no of degrees of freedom is hundreds of millions Time staggered approach permits solution of thousands of ODEs on thousands of processors We explain the time-staggerd approach first for two equations 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Staggered Partitioning in Multi-Physics Problems
Consider a fictitious two discipline model described by two scalar fields (X and Y) having two-way interaction Assume that x(t) and y(t) are governed by first order differential equations with f (t) and g(t) as known applied forces 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Solution of a Model Multi-Physics Problem
Assume that x and y at nth discrete time is known: A monolithic solution to the problem is given by backward Euler integration: This requires a repetitive solution (2x2) equations given below: 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Staggered Approach – Model Multi-Physics Problem
In staggered approach xn+1 is evaluated assuming yn+1 Here ypn+1 is a predicted value (or simply the predictor) ypn+1 can be obtained in two ways: The staggered partition approach produces simpler two equations and each of these two equations can be solved in tandem. 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Staggered Approach – Flow Chart

State Diagram of Staggered Approach
If fields X and Y are two separate programs, inter-field data transfer between them is as follows: 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Tools in Staggered-Partition Approach

Desired Characteristics of an Approach
Preserve the stability of stable mathematical models: While simulating steady fluid flows, the solution is expected to converge to a stationary state. Preserve the instability of unstable mathematical models: While simulating steady flutter or divergence, the solution is expected to diverge away when a disturbance applied Stability analysis is investigated using amplification method, which looks at decay of perturbations one time step to the next time step Accuracy. This is well defined for linear simple equations. Normally the methods accepted for linear analysis are used for non-liner problems hoping that they preserve accuracy 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Model Equations for Testing Stability
Various model equations (also called test equations are studied) with increasing complexity are studied to arrive at stability: We assume that complex equations having similar features will have the same stability: 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Stability of Forward Euler Integrator
Integrate scalar equation using multi-step method 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Stability/Over stability of Scalar Equation

Advantages of Data Transfer in Partitioned Method
In FSI, matching meshes at the fluid–structure interface is not possible because CFD requires a much finer mesh than the structure The advantages Existing CFD and FEM solvers can be reused Individual teams takes care of the meshing quality requirements of different solvers Individual CFD and FEM communities can develop advanced efficient solvers on heir own This is in contrast with the monolithic (or fully coupled) approach - the whole system of flow & structure is solved all at once Every time type of coupling requires a new dedicated solver 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Properties of Data Transfer Methods
Criteria for data exchange (or coupling method) Global conservation of energy over the interface Global conservation of loads over the interface Accuracy Conservation of the order of the coupled solvers and Efficiency (i.e ratio between accuracy and computational costs) 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Problems in Data Transfer
Partitioned coupling: non-overlapping domains for the flow and the structure Domain decomposition speeds up complex computations In FSI the meshes used in the different domains do not match at the common interface Exchange of information at the interface is non-trivial 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Transfer of Data at the Interface
FSI requires pressure loads are transmitted from the fluid side of the interface. Similarly the motion of the structure needs to be transferred to fluid side There are several methods to do the transfer for transfer of data between non- matching meshes nearest neighbour Interpolation projection methods methods based on interpolation by splines 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Summary Accurately predicting flutter speed as a function of Mach no is a challenging problem requiring High Performance Computing Multi-disciplinary analysis, such as dynamic fluid-structure interaction (aero- elasticity) is characterized by multi-scale, multi-physics, multi-processors and multi- group activity The simulation strategy needs to be defined based on efficiency of simulation rather than accuracy of simulations Closely coupled time-staggered simulations is the efficient framework for multi- disciplinary analysis Reduced order models, rather than the partial differential equations for physics should be used 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Summary Reduced order models, which are ODEs order of magnitude smaller in numbers rather than the partial differential equations for physics should be used Stability and instability of staggerd time-split approach needs to be studied for reliable results. Inter field data transfer should not destroy the underlying physics between two or more physical fields 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

References 19-Jul-2019 Fluid-Structures Interaction – HPC Perspective

Thank You ! Prof. Gopal Shevare

Fluid-Structures Interaction – HPC Perspective

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