NASA Technical Reports Server (NTRS) 20000099711: A Unified Approach to Modeling Multidisciplinary Interactions

11 Pages·0.68 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA-2000-4704
A UNIFIED APPROACH TO MODELING MULTIDISCIPLINARY
INTERACTIONS
Jamshid A. Samareh"
NASA Langley Research Center
Hampton, VA
Kumar Go Bhatia t
The Boeing Company
Seattle, WA
Abstract
There are a number of existing methods to transfer information among various disciplines. For a
multidisciplinary application with ndisciplines, the traditional methods may be required to model (n2- n)
interactions. This paper presents a unified three-dimensional approach that reduces the number of
interactions from (n2-n) to 2n by using a computer-aided design model. The proposed modeling
approach unifies the interactions among various disciplines. The approach is independent of specific
discipline implementation, and a number of existing methods can be reformulated in the context of the
proposed unified approach. This paper provides an overview of the proposed unified approach and
reformulations for two existing methods. The unified approach is specially tailored for application
environments where the geometry is created and managed through a computer-aided design system.
Results are presented for a blended-wing body and ahigh-speed civil transport.
Introduction example, the strong interactions between CSM
and CFD can prompt physically important
phenomena such as those occurring in aircraft
A key element in the application of
due to aeroelasticity. Correct modeling of these
multidisciplinary design optimization (MDO) to
complex aeroelastic phenomena requires a
an engineering system is the introduction of a
coupling of CSM and CFD for a flexible structure
consistent geometric representation. Such a
(e.g., airplane). In a multidisciplinary
representation guarantees that the same
environment, various disciplines must represent
geometry model is used to derive the
the same configuration geometry, and data from
computational models required for various
each discipline must be available consistently to
disciplinary analyses. By utilizing computer-
all the disciplines. The data may be scalar (e.g.,
aided design (CAD) for consistent geometry
pressure and temperature), vector (e.g.,
representation, it is easier to analyze complex
deflection and heat transfer), or integrated
configurations with higher-fidelity tools such as
quantities (e.g., aerodynamic and thermal
computational fluid dynamics (CFD),
loads). The data transfer process may be
computational structural mechanics (CSM), or
subjected to additional constraints, such as
detailed finite-element analysis.
conservation of forces, moments, and energy.
The focus of this paper is the transfer of data
A feature that characterizes multidisciplinary
between dissimilar grids (most models do not
analysis and optimization is the modeling of
share the same nodal locations at the interface).
interactions among various disciplines. For
"Research Scientist, Multidisciplinary Optimization Branch, AIAA Senior Member
,Senior Technical Fellow, Aeroelasticity and Optimization, AIAA Associate Fellow
American Institute of Aeronautics and Astronautics
AIAA-2000-4704
In the recent years, various researchers have
examined the issue of aeroelastic transfer.
Review of Existing Methods Discrepancies and dissimilarities in geometry
and grid models are two potential sources of
error. The accuracy of the data transfer depends
The issue of exchange of pressure distribution
on the relative resolutions of disciplinary grids.
and displacements between an aerodynamic
Data could be lost in transfer from a coarse grid
model and a structural model gained urgency
to a fine grid. Another source of error occurs if
with the first widespread use of panel
the models have dissimilar levels of geometry
aerodynamic methods and finite-element
detail. As noted by Tzong et al.,4 a CFD grid
structural models during the late sixties. The
generally resembles the true geometry of the
traditional beam and strip theory methods were
aircraft; the grid includes details such as pylons,
not applicable to the large, flexible low-aspect
nacelles, flaps, and slats. However, a CSM grid
ratio configuration of a supersonic transport
generally represents only major structural
(SST). The SST experience led to development
components, such as the wing box. Flaps and
of a comprehensive aeroelastic computer
slats are represented either by a few simple
program called FLEXSTAB. 1 Most of the work
beam elements or are completely excluded.
was done under NASA funding before 1974. The
Tzong et al.4and Kapania and Bhardwaj s have
basic theoretical work was completed between
developed methods, based on finite element
1968-71, and the goal of FLEXSTAB was to
(FE) technology, in which virtual work is
provide an aeroelastic program for production
employed to transfer the aerodynamic pressures
use. FLEXSTAB probably was the first
onto a CSM grid. The displacements are then
computing system to address systematically all
converted back to a CFD grid through the
the aeroelastic transfer issues for a complete
reciprocal theorem. Kapania and Bhardwaj s
airplane configuration. FLEXSTAB was
were successful in using a simplified version of
successful in providing an acceptable method
this method for several wings.
for the pressure and displacement transfers.
However, the method was time consuming, and
Brown6 added virtual elements in the CSM
required significant manual effort for the user to
model to cover the discrepancies in the
establish direct correspondence between the
geometry definition between CSM and CFD
aerodynamic panels and structural elements
models. These virtual elements add neither
stiffness nor mass. As pointed out by Cebral and
In the eighties Dassault Aviation developed a
LShner, 7the generation of the virtual elements
more simplified method2'3 including many
is an unnecessary complication, particularly for
practical considerations. The Dassault method is
embodied in Elfini and includes several complex geometries.
innovative ideas in aeroelastic formulation. First,
Hounjet and Meijer8 and Smith, Hodges, and
Dassault used an intermediate computational
Cesnik9provided overviews of the data transfer
grid between an aerodynamic mesh and a
methods. Smith, Hodges, and Cesnik 9evaluated
structural finite-element model for a given
six methods for transferring information between
configuration. Thus different aerodynamic
models could be used with the same structural CFD and CSM disciplines. These methods were
infinite-plate spline (IPS), multiquadric
model, and vice versa. Second, they used shape
biharmonic (MQ), nonuniform B-spline (NUBS),
functions defined on the computational grid, and
thin-plate spline (TPS), finite-plate spline (FPS),
smoothing operators to relate structural
and inverse isoparametric mapping (IIM). These
displacements to the aerodynamic mesh. Third,
methods have been implemented in a single
they introduced the idea of loads basis on the
code, FASIT _°.The IPS method is based on the
computational grid (unit loads at the nodes of
popular surface splines11and is implemented in
the computational grids) to transfer pressure
some commercial aeroelastic analysis tools.
from the aerodynamic mesh to the
This method is designed for interpolating a
computational grid, and from the computational
function of two variables.
grid to the finite-element nodes. The overall
approach is elegant and practical. But it still
Out of these six methods, Smith, Hodges, and
requires significant user input (although much
Cesnik9 recommended further study of IIM and
less than FLEXSTAB) and checking to ensure
NUBS. They indicated that IIM shows great
the accuracy of the transfers.
promise for two-dimensional applications and
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AIAA-2000-4704
needed to be extended to three dimensions. proposed by Murti _9 for a two-dimensional
Clutter 12and Send13extended NUBS to three model; it uses the FE shape functions to
dimensions. interpolate the coordinates, pressure, and
displacement vectors. The method does not
One problem found with the NUBS require any matrix inversion. Because the FE
implementation is that the data must be input as shape functions satisfy a positivity constraint,
a structured (regular) grid. This requirement the process will not create nonexisting local
forces the data, at best, to be approximated, and extrema. The local interpolation is computed by
in most realistic cases, this step is either time projecting one grid onto another. Cebral and
consuming or impossible. Samareh proposed L6hner7'_ presented a variation of the IBA that
TM
a method to use non-uniform rational B-spline could guarantee the conservation of forces.
(NURBS) representation for data transfer among They used a Galerkin method to solve for
various disciplines. Because this method is pressure from the CSM grid. The CebraI-L6hner
based on a general three-dimensional, least- method requires a matrix inversion. They also
squares representation, 1"s'_6it does not require used an adaptive Gaussian integration
the input to be a structured grid. Another technique to improve the accuracy. Farhat,
advantage of this approach is control over the Lesoinne, and LeTallec 21 also presented a
tradeoff between smoothness and accuracy. variation of the original IBATMthat can guarantee
conservation of forces, but requires no matrix
The accuracy of the data transfer process for inversion. We used this algorithm for the current
integrated quantities (e.g., forces, moments, and study.
energy) depends on the consistency of data
transfer as well as other constraints, such as Unified Avvroach
conservation. For example, a consistent load
vector for CSM isdefined as
Overview
{],}=_{N}TE{P}dS The unified approach has two essential
ingredients. First, the data transfer process
S
between two disciplines was modeled by a
transformation matrix. Second, the CAD model
where {]e } is the element load vector, {N} is was used to reduce the number of interactions.
the FE shape function, K is the unit surface
normal, P is the pressure, and dS is the The interaction between two disciplines is
infinitesimal surface element. modeled mathematically as
{F2}=[T2,]{F,}
Because the above equation uses the same
shape functions as are used to calculate the
element stiffness matrix, the equation
where matrices {F I} and {F 2}contain the
guarantees a consistent loading or lumping.
Cook, Malkus, and Plesha _7provided a detailed information on discipline grids 1 and 2,
description of this equation. There are several respectively, and matrix [T21] is a
possible problems with using the above
transformation matrix. For example, {F 1}could
equation. First, the aerodynamic load may have
be an aerodynamic loads vector defined on a
a large variation within a single FE, such that the
shape function is not adequate to capture the CFD grid and transferred to the CSM grid as
variation. Second, the FE shape function may {F2}. Generally the transformation matrices
not be available for some commercial CSM are sparse and large. Only the nonzero
codes. Third, in its present form the above elements need to be stored. Ifthe transformation
equation does not guarantee conservation of
matrix [T2t] is independent of the shape
forces and moments.
changes, then [T21] can be calculated once
Maman and FarhatTM outlined a consistent and used as long as there is no change in the
interpolation-based algorithm (IBA) for grid connectivity.
transferring information between two dissimilar
grids. The algorithm is similar to the IIM The concept of a transformation matrix simplifies
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integrated analyses such as aeroelastic the fact that the sensitivity derivatives used in
calculation. The aeroelastic calculation has four
gradient-based optimization with respect to a
distinct steps. First the aerodynamic loads are
vector of shape design variables {Vi} will not
calculated on the CFD grid. Second the loads
require the differentiation of the transformation
are transferred to the CSM grid. Third the
aeroelastic deflections are calculated on the matrix. For example, the following relation could
CSM grid. Fourth the deflections are transferred transfer the sensitivity of the CFD load to the
CSM grid:
to the aerodynamic grid to recalculate the
aerodynamic loads. This iterative process can
be expressed as
{.,_s}= {Flow Solution (G-F+ 6"F)}
For the six methods described in Ref. 9, the
[K]{s,}{=T,} transformation matrices are dependent directly
on the shape changes. However, NURBS and
IIM methods can be reformulated to result in
methods with transformation matrices that are
independent of the shape changes. Samareh
TM
The first equation represents the aerodynamic proposed a reformulation for NURBS, and
load calculation. The term Gr represents the Maman and FarhatTM proposed a general
alternative approach for IIM. This paper provides
CFD grid, and 8F is the aeroelastic deflection reformulations for both the methods.
on the CFD grid. The aerodynamic forces, Fr,
The second ingredient of the proposed unified
are transferred to the CSM grid as Fs. The approach helps to reduce the number of
matrix [K] is the CSM stiffness matrix, and transformation matrices. For a multidisciplinary
application that involves ndisciplines, the
_'sisthe aeroelastic deflection on the CSM grid.
traditional process may require (n 2-n)
transformation matrices. However, some of
The use of the transformation matrix simplifies
these couplings are either weak or nonexistent.
the above set of equations to
Figure 1 shows all possible interactions among
[K]{s}=, eight disciplines; modeling all interaction
requires 56 transformation matrices.
The problem can be further complicated for a
variable-fidelity multidisciplinary application. For
For a linear structure without rigid body degrees- example, aerodynamic loads can come from
of-freedom, the above equation set can be wind tunnel databases, or from linear
simplified to aerodynamics, potential flow, Euler, and Navier-
Stokes analysis codes. The following section
provides details of an approach where the
{(_F}= [Trr ]{Fs};where
number of couplings is reduced from (n 2-n)
[Trr]=[TsslK]-l[Tss] to 2n by incorporating a CAD model.
The formulation can be extended to the case
with rigid body degrees-of-freedom, and itallows CAD-Based Approach
decoupling of the CFD code from a linear CSM
code.
The new element of the CAD-based approach
was the use of a CAD geometry representation
Ifthe transformation matrix is independent of the to reduce the number of transformations from
shape changes, then the formulation is
(n 2-n) to 2n. The reduction is accomplished
especially beneficial. The benefit results from
by transferring the data to a CAD geometry
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model that serves as a common bridge or adata I 1
bus for sharing information among various
disciplines, as shown in the Fig. 2. m i )
R(u,v) =
I J
This approach is of obvious benefit for
multidisciplinary applications with more than i j
three disciplines. However, the method offers
benefits even when only two disciplines are
where the _ are the locations of NURBS
involved. The intermediate CAD grid isolates
each discipline model from changes to all other control points, the Wqcoefficients are the
discipline models. The overall approach has a
weights, u and v are the parameters,
strong potential for robust automation of all
transformations, and ensures consistency l andJare the numbers of control points in
among different renderings of the same
configuration. u and v directions, and Bi.p(U ) and BLq(V )
are the B-spline basis functions of degrees
First the data is transferred from the individual p and q respectively. This equation can be
source discipline to the CAD model as written in a compact form by combining the
weights and basis functions into asingle term as
{Fc}=[rc],{F},
I*J
v):Z c.(u, =
Then the data is transferred from the CAD to the
target disciplines. n
where n=i+I*(j-1), and
{F.}=
WiiBi, p(u)Bj,q (v)
C. (u,v) = (u, v) =
1 J
ZZWktB,.p(U)B,.q(v)
The advantage of this approach is that only k 1
[Tc,] and [TEc] transformation matrices have to
be calculated: 2n matrices instead of (n2- n). Typically the data from the source discipline,
= {fi,f2 ..... J_m.... }r, defines loads or
Transformation Matrices
deflection vectors at a discrete set of points 7m.
In the first step we project each source grid point
It is possible to reformulate the existing methods
in terms of transformation matrices. This section Ymonto the CAD NURBS surfaces and find the
presents reformulations of two existing appropriate surface and the associated
alternative methods.
parameters, u,,and v,,. The projection reduces
the number of independent variables from three,
NURBS-Based Interface 14
The NURBS-based process has two steps. In J'm(Xm,Ym,Zm), to two, _f,.(u,.,v,). The
the first step, the discipline model is mapped to
independent variables (u,,, v,, ) are the
the CAD model. Then in the second step, a
NURBS representation is used to fitthe data. parametric coordinates of the point 7m on the
NURBS surface. This information may be
Most CAD systems provide tools to save a CAD
available from the grid generation process. If
model as a NURBS representation, which then
not, the grid points can be projected onto the
can be used for the mapping step. This section
original NURBS surface (see Ref. 23).
contains a brief overview of NURBS
representation; readers should consult Refs. 14
A NURBS surface is then fitted through the data
and 22 for more detailed discussion. A NURBS
to form
surface, R(u, v), can be represented as
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Interpolation-Based Algorithm
We followed the algorithms proposed in Refs. 18
and 21. The first step is to map each source and
The fitted surface is based on a least-squares target grid point to the CAD model as described
approximation (see Refs. 14-16) that minimizes earlier, and map the CAD model to the source
the approximation error. The weights and the and target disciplines. We used a discrete
knot vectors of the CAD NURBS surface can be
representation of the CAD model, which is
used, or the user can specify weights and knot readily available from most CAD systems. Each
vectors. The degrees of NURBS approximation, source and target point was mapped to the CAD
p and q, and the knot vector distribution affect model, and the parametric coordinates (u, v) of
the smoothness of the least-squares the source and target grid points were
representation. The minimization error E can determined from the mapping process. The
be written as second step is to transfer the data from the
source discipline to the target discipline. In this
: = step, the parametric coordinates were used to
transfer the data tothe CAD model:
{{r,}-[c]{Tc}K
Fc = Z N,(ui,,vi,)_,
i
The least-squares form of the above equation
can be expressed in a matrix form as
where the termN i was the FE shape function.
This equation was applied for each source grid
point. The resulting set of equations was
where assembled into
{rc}=[rc,
A generic high-speed civil transport (HSCT)
The elements of the transformation matrix were
geometry was used to demonstrate the
made of the FE shape functions. A similar
algorithm 14.This geometry was made of three
process was used to transfer the data from the
surfaces: fuselage, inboard wing, and outboard
CAD model to the target discipline. As
wing. Figure 3 shows the original (undeflected)
demonstrated by other researchers, 7:a the IBA
NURBS surfaces, the deflected CSM grid, and
the deflected NURBS surfaces. To test the was very effective in transferring the scalar and
limits of the deflection-transfer algorithm, the vector quantities.
CSM grid had a large and unrealistic deflection.
The original and the modified IBA2_were used
TM
to transfer the aerodynamics data for a blended-
Because the CSM grid is generally coarser than
the CFD surface grid, the interpolation approach wing body and an HSCT model. Figure 4 shows
the result of transferring sensitivity derivative
for deflection transfer may produce a
data from aCFD grid to a CSM grid for an HSCT
discontinuous CFD surface grid. However, the
model using the original algorithm. The data is
NURBS-based approach maintained the
the sensitivity derivative of pressure with respect
smoothness of the geometry for deflection
to the leading-edge sweep angle.
transfer. This is a major advantage for the
NURBS based approach. On the other hand, for
Figure 5 shows the result of transferring the
transferring the integrated quantities, the
process in its present form did not guarantee pressure distribution for a blended-wing body
conservation of forces and moments. Further using the original algorithm. The figure shows
studies need to be performed to determine the pressure contours on the CFD grid and the
whether the advantages from the smoothness of pressure contours transferred from the CFD grid
the NURBS approach outweigh the lack of to the CAD model. It also shows the pressure
contours transferred from the CAD model to the
conservation.
CSM grid. As expected, the original IBAl° did not
maintained the conservation of forces and
moments. The error for the integrated forces
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was approximately 5%. This error would be dimensional data transfer. The winglet is not
higher when the models have dissimilar levels of merely shifted up--it has followed the CSM
geometry details. deflection without neglecting the x and y
components of the deflection. The traditional
For transfer of the integrated quantities, the data transfer will only include the vertical
modified IBA had a slightly different
displacement (z coordinates).
implementation (see Ref. 21), which conserves
forces and moments. Figure 6 shows the load
Summary
vectors on the CFD grid and the load vectors
transferred from the CFD grid to the CAD model.
It also shows the load vectors transferred from We have presented a unified approach for
the CAD model to the CSM grid. As expected 21, transferring information for a multidisciplinary
application with n disciplines. This method has
the transfer process maintained the
conservation of forces and moments within the two essential ingredients. First, the data transfer
process between two disciplines was modeled
machine's accuracy. This process can also be
used to transfer deflections from a CSM grid to a by a transformation matrix. Second, the CAD
model with consistent geometry was used to
CFD grid.
reduce the number of interactions from
Figure 7 shows the result of transferring the (n2-n) to 2n. This unified approach was
aeroelastic deflections from a CSM grid to a specially tailored for application environments
CAD model and then to the CFD grid. The top- where the geometry is created and managed
left figure shows the deflected CSM grid. The through a CAD system. Results were presented
deflection is transferred from the CSM grid to the for a high-speed civil transport and a blended-
CAD model, as shown in Fig. 7 (middle-left wing body. The approach provides a framework
figure). Then the deflection is transferred from where data transfer among disciplines can be
the CAD model to the CFD grid (bottom-left accomplished consistently, and potentially with a
figure). The right portion of the figure shows the high degree of automation. Comparisons
deflected CSM and CFD grids, and they are between the NURBS and IBA using CAD and
right on top of each other. A closer look at the transformation matrices are continuing.
Fig. 7 demonstrates the benefit of the three-
Discipline 1 <3<_,0_//_
%
E_
o_.
-a
g eu!ld!0s!G
Fig. 1Multidisciplinary interactions.
Fig. 2Unified multidisciplinary interactions.
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Deflected Deflected
CSM Gr_ CAD Surface
Original Original
CSM Surface CAD Surface
Fig. 3 Aeroelastic deflection transferred from CSM grid to a CAD model.
!
,_ CFD ._
CAD
CSM ,.
Fig. 4 Transfer of sensitivity derivative from CFD grid to CSM grid.
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CFD
CSM CAD
Fig. 5 Pressure interpolation from CFD grid to CSM grid.
CFD
CSM
Fig. 6 Transfer of aerodynamic load vectors from CFD grid to CSM grid.
American Institute of Aeronautics and Astronautics
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CSM
CSM and CFD
CAD
CFD
Fig. 7 Transfer of aeroelastic deflection from CSM grid CFD grid.
References
1Dusto, A. R., "A Method for Predicting the Stability Characteristics of an Elastic Airplane, FLEXSTAB
Theoretical Description," NASA CR-114,712, Oct. 1974.
2 Petiau, C. and Brun, S., "Trends in Aeroelastic Analysis of Combat Aircraft," AGARD, AD-P005855,
Aug. 1987.
3 Nicot, Ph., and Petiau, C., "Aeroelastic Analysis Using Finite Element Models," European Forum for
Aeroelasticity and Structural Dynamics," Aachen, Germany, 1989.
4Tzong, G., Chen, H. H., Chang, K. C., Wu, T., and Cebeci, T., "A General Method for Calculating Aero-
Structure Interaction on Aircraft Configurations," AIAA Paper 96-3982, Sept. 1996.
5Kapania, R. K., and Bhardwaj, M., "Aeroelastic Analysis of Modern Complex Wings," AIAA Paper 96-
4011, Sept. 1996.
6Brown, S. A., "Displacement Extrapolations for CFD+CSM Aeroelastic Analysis," AIAA Paper 97-1090,
Apr. 1997.
z Cebral, J. R., and L6hner, R., "Fluid-Structure Coupling: Extensions and Improvements," AIAA Paper
97-0858, Jan. 1997.
8Hounjet, M. H. L., and Meijer, J. J., "Evaluation of Elastomechanical and Aerodynamic Data Transfer
Methods for Non-Planar Configurations in Computational Aeroelastic Analysis," The Proceedings of the
American Institute of Aeronautics and Astronautics