[NASA美国太空总署资料（长期更新）].NASA-aiaa-96-3991.USE.OF.CAD.GEOMETRY.IN.MDO

AIAA-96-3991 USE OF CAD GEOMETRY IN MDO Jamshid A. Samareh j.a.samareh@larc.nasa.gov Computer Sciences Corporation, Geometry Laboratory (GEOLAB, http://geolab5.larc.nasa.gov) (ftp://techreports.larc.nasa.gov/pub/techreports/larc/96) 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization September 4-6, 1996 Bellevue, WA 1 USE OF CAD GEOMETRY IN MDO � Jamshid A. Samareh y 1 Abstract The purpose of this paper is to discuss the use of Computer-Aided Design (CAD) geometry in a Multi- Disciplinary Design Optimization (MDO) environ- ment. Two techniques are presented to facilitate the use of CAD geometry by di�erent disciplines, such as Computational Fluid Dynamics (CFD) and Compu- tational Structural Mechanics (CSM). One method is to transfer the load from a CFD grid to a CSM grid. The second method is to update the CAD geometry for CSM de ection. 2 Introduction The process of aircraft design can be broken into three phases[1]: (1) conceptual design, (2) prelimi- nary design, and (3) detail design. The conceptual design process focuses on the basic design optimiza- tion of features, such as weights, sizes, and overall performance. During the preliminary design, the fo- cus is on the mathematical modeling of the outside skin of an aircraft with su�cient accuracy. After this phase, the geometry is frozen, and any change could be costly. Detail design concentrates on the actual design of pieces to be fabricated. Often an aircraft is represented by a simple model during the conceptual and preliminary designs. Be- cause simple models are neither accurate nor com- plete, optimization of these models could lead to an impractical design [2, 3]. This shortcoming can be alleviated by using a high �delity model, and the in- teraction among various disciplines must be modeled accurately. These interactions are very complicated and important piece of MDO. The strong interactions of CSM and CFD are very common in an MDO environment. Such interactions can prompt physically important phenomena such as those occurring in aircraft due to aeroelasticity. Cor- � This paper is declared a work of the U. S. Government and is not subjected to copyright protection in the United States. (ftp://techreports.larc.nasa.gov/pub/techreports/larc/96) y Senior Computer Scientist (j.a.samareh@larc.nasa.gov), Computer Sciences Corporation, Geometry Laboratory (GE- OLAB, http://geolab5.larc.nasa.gov) rect modeling of these complex aeroelastic phenom- ena requires direct coupling of CSM and CFD. Dur- ing optimization of a exible structure (e.g., wing), the geometry changes due to the aeroelastic e�ect. All disciplines share the same geometry, and must be able to consistently communicate and share informa- tion (e.g. de ection and load). The geometry repre- sentation for MDO must be accurate and suitable for grid generation for various disciplines such as CFD and CSM. To further complicate the process, model- ing of complicated geometrical models requires use of CAD systems. The interactions among various disciplines require the manipulation of the original CAD geometry that is stored as a set of NonUniform Rational B-Splines (NURBS). This paper describes two techniques to manipulate the NURBS geometry. In the next fol- lowing sections there are brief discussions on NURBS, parameterization of aircraft geometry, NURBS-based optimization, load transfer, de ection transfer, re- sults, and conclusions. 3 NURBS This section contains a brief overview of the NURBS, and readers should consult [4] for a detailed discus- sion. A NURBS curve, ~ R(u), can be represented as ~ R(u) = P n i=0 N i;p (u)W i ~ P i P n i=0 N i;p (u)W i : (1) The parameter, u, is bounded by u min � u � u max . The ~ P i are the control points (forming a control poly- gon), and W i are the weights. The N i;p are the p-th degree B-spline basis functions de�ned on the non- periodic and nonuniform knot vector (u) u 2 [u min = u 0 � u 1 � . . .u j � . . . � u k = u max ]; (2) where k is the number of knots. This completes the mapping between the one-dimensional parameter space, u, and the three-dimensional Euclidean space, ~ R. A NURBS curve has �ve important properties: � It is invariant under linear transformation. 2 � A NURBS curve of order p, having no multiple interior knots, is p - 2 di�erentialable. � The approximation is local in nature. � A NURBS curve is contained in the convex hull of its control points. � The NURBS approximation is variation dimin- ishing. To evaluate, the three-dimensional curve NURBS is commonly represented in homogeneous form as ~ R(X;Y; Z)() ~ R W (WX;WY;WZ;W ): (3) So, the NURBS curve can conveniently be de�ned as a perspective map of its nonrational counterpart in four-dimensional space as ~ R W = n X i=0 N i;p (u) ~ P W i ; (4) where ~ P W i is de�ned as ~ P W i = fX i W i ; Y i W i ; Z i W i ;W i g: (5) The basis functions can be e�ciently computed by using DeBoor algorithm [5] as N i;0 (u) = � 1 if u i � u < u i+1 ; u i < u i+1 , 0 otherwise (6) N i;p (u) = L i (u)N i;p�1 (u) +M i (u)N i+1;p�1 (u); (7) where L i (u) = u� u i u i+p � u i ; (8) M i (u) = u i+p+1 � u u i+p+1 � u i+1 : (9) It is agreed that 0=0 = 0. A NURBS surface is a parametric surface and is de�ned as a function of two parameters as ~ R( ~ U ) = fX( ~ U ); Y ( ~ U ); Z( ~ U )g T ; ~ U = fu 1 ; u 2 g T 2 [(a; b); (c; d)]; (10) where the components of vector, ~ U , are the sur- face parameters and have no geometrical signi�cance. However, for a constant u 2 , as u 1 increases the point, ~ R( ~ U ) moves always from one side of the surface to the other side. The NURBS de�nition for the surface is de�ned as ~ R( ~ U ) = P n i=0 P m j=0 N i;p (u 1 )N j;q (u 2 )W i;j ~ P i;j P n i=0 P m j=0 N i;p (u 1 )N j;q (u 2 )W i;j : (11) where ~ P i;j are control points (forming a control sur- face), W i;j are the weights, and N i;p and N j;q are the p-th and q-th degree B-spline basis functions de- �ned on the non-periodic and nonuniform knot vec- tor. The evaluation process is very similar to the NURBS curve evaluation. 4 Aircraft Parameterization To use complex shapes in an MDO environment, the parameterization and geometry modeling must be compatible with existing CAD systems, and it must be adaptable to CFD (i.e., block-structured or un- structured grids) and CSM. The four approaches for parameterization of an aircraft geometry are based on: analytical, semi-analytical, discrete, and CAD representation. Analytical methods for optimization have been used for a long time. These methods converts a set of design variables (e.g., wing sweep, thickness ra- tios) into a set of surfaces. Then, these surfaces can be used to analyze and compute the objective func- tion (e.g., [6, 7, 8]). This approach is very simple and requires a few design variables. The geometry based on this approach is very smooth. On the other hand, the geometry can take a limited form, and it is hard to optimize existing and free-form geometry. Inter- actions among disciplines are very di�cult to model. Blair and Reich [8] have implemented a Virtual De- sign Process (VDM) that is integrated with full as- sociativity within Pro/Engineer CAD/CAM software [9]. In the second approach, semi-analytical, a set of points can describe the initial geometry, and a poly- nomial can model the perturbation of geometry [10]. Then, the coe�cients of this polynomial are used as a set of design variables. Again, this method is simple, and it allows the designer to use existing geometry. This approach requires a few design variables, and the smoothness of the geometry depends on the baseline geometry. This approach is very di�cult to general- ize and use in an MDO environment for a complex geometry. The third approach, discrete, is based on a discrete representation of the geometry. The baseline geom- etry creates the grids, and the position of each grid point becomes a design variable for the optimizer. 3 This is very easy to implement, and the geometry changes don't have a limited form. The latter could create a problem in which the optimum design may not be practical to manufacture. The number of de- sign variables often becomes very expensive which, leads to high costs and a di�cult optimization prob- lem to solve. Also it is di�cult to maintain a smooth geometry. The fourth approach, CAD, is based on the op- timization of a NURBS representation of geometry (e.g., [11]). The large number of design variables of- ten becomes very expensive which leads to high costs and a di�cult optimization problem to solve. The geometry continuity and smoothness are guaranteed. Also, the geometry can change locally without a�ect- ing everything else. This type of parameterization is exible enough to represent a wide range of geome- tries. Existing complicated CAD models can be used as the baseline model, but modeling the interaction among disciplines is very di�cult. These four approaches are summarized and listed in Table 1. In the next section, a NURBS-based opti- mization that is a hybrid approach based on �rst and fourth approaches is discussed. 5 NURBS-Based Optimization CAD systems have been developed very rapidly and integrated into the design process(e.g., see [8]). Use of CAD systems for geometry modeling in an MDO en- vironment could potentially save development time. However, there are two drawbacks: (1) initial invest- ment (software and training), and (2) inability to calculate analytical sensitivity. The geometry rep- resentation in these systems is complicated due to a large number of entity representations. In a tra- ditional CAD system, the geometry is represented as one of many possible mathematical forms such as Bezier, Coons patch, B-Spline, surface of revolution, etc. However, one can use NURBS equations to rep- resent most parametric and implicit surfaces without loss of accuracy [4]. NURBS can represent quadric primitives (e.g., cylinder, cones), as well as free form geometry [4]. There are some CAD surfaces (e.g., helix and helicoidal[12]) that cannot be directly con- verted to NURBS. These surfaces are not common in most CAD system. The NURBS representation must be used in such as a way that it should be possible to altered it automatically to accommodate the changes due to: (1) design variable changes, or (2) structural and control surface de ections. Calculating of the sensitivity of geometry with re- spect to the design variables could prove to be very di�cult. In some instances, it is possible to relate the NURBS control points to the design variables. Then the analytical sensitivity can be calculated out- side of the CAD system. Another way to calculate the sensitivity is to use �nite di�erence, as long as the perturbed geometry has the same topology as the unperturbed one. Both methods, the analytical and �nite di�erence, have their pitfalls and limitations. Implementation described here is based on the Framework for Interdisciplinary Design Optimization (FIDO) [13, 14] developed at NASA Langley Re- search Center (LaRC). The process for geometry cre- ation, integration, and manipulation are designed around the NURBS representation of the complete geometry. To embed this process into an optimiza- tion process such as FIDO, the model (High-Speed Civil Transport) has to be parameterized with a set design variables (DVs). For each optimization cycle, the run starts with a de�nition for a set of design variables (see Figure 1), to build a NURBS-based geometry. Pro-Engineer ([9]) is proposed for this step as the geometry builder. Once the geometry is built, the NURBS geometry will be deposited into a NURBS database, which will be shared among di�erent disciplines. This database will be maintained as the baseline geometry changes. During the optimization process, CFD and CSM dis- ciplines will need the complete geometry de�nition. For each iteration of the aeroelastic loop, the CAD geometry is used to create CFD and CSM grids. This requires that both disciplines use and modify the same unique geometry in the geometry database. In each loop, the CFD grid is used to compute the aero- dynamic load, which is transferred to the CSM grid. This load will be converted to a NURBS de�nition and deposited into the database. At this point, the CSM grid will be generated based on the NURBS database. The NURBS CFD load will be mapped to the CSM grid, which along with the CSM grid is used to compute the resulting de ection. The next criti- cal step is to modify the original CAD geometry to accommodate and re ect the aeroelastic de ection. Each optimization loop can be summarized as shown in Figure 1: � 1. The CAD system, Pro/Engineer, will convert a set of design variables (D) into a NURBS geom- etry (G). Because the source for Pro/Engineer code is not available, the sensitivity has to be 4 calculated outside of the system. � 2. At this point, the geometry, G, and its sensitivity, GD, will be stored in the NURBS database. � 3. The Coordinates and Sensitivity Calcula- tor for Multidisciplinary Design Optimization (CSCMDO) ([15]) will calculate the CFD mesh and its sensitivity M, MD. � 4. The CFD code, with the capability to calcu- late the sensitivity, will obtain the load, which will be stored in the NURBS database. � 5. Similarly, the CSM grid generator can cre- ate the �nite element grid and its sensitivity. The load stored in the NURBS database will be mapped onto the �nite element model. � 6. The CSM code, with the capability to cal- culate the sensitivity, will obtain the de ection which will be stored in the NURBS database. � 2'. The �nal step is to modify the NURBS database to accommodate the new structural de- ection. The two steps that are de�ned and discussed in this paper in detail are: (1) converting CFD load to a NURBS de�nition, and (2) modify the NURBS geometry to accommodate the de ection. 6 Load Transfer To transfer the load to CSM elements, one needs to be able to compute the load at any point on the sur- face. Initially, the load, F ( ~ R) CFD , is computed on the CFD grid, ( ~ R CFD = fX;Y; Zg T ), which could be a set of structured quadrangles or unstructured tri- angles with the appropriate connectivity. There are two basic problems in �tting this data with NURBS: (1) the data has four dimensions, (X;Y; Z; F ), and (2) the CFD grid could be an unstructured grid. The �rst problem can be solved by mapping the CFD grid to the original NURBS surfaces, hence, reducing the dimension from four, (X;Y; Z; F ), to three, (U; V; F ). The U; V are the parametric coor- dinates of the original NURBS surface. This infor- mation may be available from a CFD grid generation process. If not, the CFD grid points can be projected onto the original NURBS surface [16]. The process of projecting a point, ~r = fX;Y; Zg T , on a surface, ~ R( ~ U ), can be performed by �nding a ~w such that the distance, d, between the ~r and ~ R(~w) is minimal and ~w is constrained to 2 [(a; b); (c; d)]. The distance, d, can be written in terms of parameters ~w as d 2 (~w) = f(~w) = j ~ R(~w)� ~rj � j ~ R(~w)� ~rj: (12) The next step is to �tting a single-value three- dimensional surface, F = F (U; V ). This surface can be �tted based on a least-squares approximation [17, 18] that minimizes the approximation error. A three-dimensional curve is used as an example of the least-squares �tting. A set of points in three space, ~r(u), can be �tted by a B-Spline curve, ~ R(u). The B-Spline equation can be expressed at each parameter, u, as ~r j = N 1;k (u j ) ~ R 1 +N 2;k (u j ) ~ R 2 + :::+N n;k (u j ) ~ R n (13) The above equation can be expressed as [~r] = [N ][ ~ R] (14) where [~r] T = [~r 1 ; ~r 2 ; :::; ~r jmax ] T (15) [ ~ R] T = [ ~ R 1 ; ~ R 2 ; :::; ~ R n ] T (16) [N ] = 2 4 N 1;k (u 1 ) ::: N n;k (u 1 ) N 1;k (u j ) ::: N n;k (u j ) N 1;k (u jmax ) ::: N n;k (u jmax ) 3 5 (17) where jmax is the maximum number of data points, ~r j . ~ R n is the nth B-Spline control points. If jmax = n, the matrix [N ] is a square matrix and the control points can be calculated directly by matrix inversion, [ ~ R] = [N ] �1 [~r]: (18) In this case, the resulting B-Spline curve passes through each data point. However, if the number of data points, jmax, is greater than the number of control points, n, the problem is over-speci�ed. A least-square method can solve the problem as, [ ~ R] = [[N ] T [N ]] �1 [N ] T [~r]: (19) The least-squares approximation for surfaces is very similar to the least-squares approximation for the curves. The minimization error can be written as 5 Error = P N 0 E N ; E N = � F ( ~ U N )� P n i P m j N i;p (U)N j;q (V )W i;j F i;j P n i P m j N i;p (U)N j;q (V )W i;j � 2 (20) where F i;j are control points for the NURBS sur- face representing the CFD load,W i;j are the weights, and N i;p and N j;q are the p-th and q-th degree B- spline basis functions de�ned on the non-periodic and nonuniform knot vector. N is the number of points in the CSM grid. This forms a system of linear equations that can be solved for control points of a NURBS surface representing the load. 7 De ection Transfer As described in the previous section, the load is de- �ned on the CFD grid, ~ R CFD = fX;Y; Zg T . On the other hand, the de ection, � ~ R CSM , is de�ned on the CSM grid, ~ R CSM = fX;Y; Zg T , which is represented by a set of polygons (e.g., triangles and quadrangles) with appropriate connectivity. The goal is to modify the CAD geometry de�nition, ~ R( ~ U ), such that it re- ects the de ection produced by CSM. The algorithm for de ection transfer has four steps: 1. Project the ~ R CSM onto the original NURBS sur- face. 2. Create a NURBS surface based on the de ection, � ~ R CSM (U; V ), which has the same degree as the original NURBS surface. 3. Add/remove knots from the new surface to make it compatible with the original NURBS surface. 4. Add the control points to the original NURBS surface to form the new surface. This algorithm has following properties: 1. As � ~ R CSM approaches zero, the method will re- produce the original NURBS surface, ~ R( ~ U ). 2. Smoothness is controlled on the resulting NURBS surface. 3. The results surface is a NURBS surface with the same degree as the original NURBS surface. 4. It is possible to maintain the same knot vector as the original NURBS surface. 8 Results and Discussions The results are presented for the load and de ection transfers. For the load transfer, a generic path�nder geometry (see Figure 2) is used. The geometry is a single cubic NURBS surface with 53 by 24 control points. There are two test cases for the load transfer. For the �rst test case, a Sin function is de�ned over a triangular mesh (see Figure 3), which covers the surface of the path�nder. The data is �tted with a NURBS surface. The original and interpolated con- tours are shown in Figure 4. The Root Mean Squares (RMS) error for interpolation is less than two per- cent, and the resulting cubic NURBS surface has 15 by 15 control points. For the second test case, the pressure distribution on the surface is �tted with a cubic NURBS surface (see Figure 5). The resulting NURBS surface has 35 by 24 control points, and the