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Engineering Design 11 -
Several parameters have been used to correlate fatigue crack growth rates under mixed-mode condi- tions such as equivalent stress intensity factors, equivalent strain intensity factors, strain energy density, and the J -integral. Fatigue cracks can also deflect in new directions and branch into multiple cracks under mixed-mode loading conditions. References 2 and 3 discuss these complex fatigue crack growth issues in detail.
References
- Collins, J. 1991. Failure of Materials in Mechanical Design , 2nd ed., Wiley Interscience.
- Suresh, S. 1991. Fatigue of Materials. Cambridge University Press, Cambridge, U.K.
- Stephens, R.I., Fatemi, A., Stephens, R.R., and Fuchs, H.O. 2001. Metal Fatigue in Engineering , 2nd ed., Wiley Interscience, New York.
- Norton, R.L. 1996. Machine Design. Prentice Hall, Englewood Cliffs, NJ.
- Stouffer, D.C. and Dame, L.T. 1995. Inelastic Deformation of Metals: Models, Mechanical Properties, and Metallurgy. Wiley Interscience, New York.
- Dowling, N.E. 1993. Mechanical Behavior of Materials. Prentice-Hall, Englewood Cliffs, NJ.
- Progress in Measuring Fracture Toughness and Using Fracture Mechanics, Materials Research Standards , ASTM (March 1964): 103–19.
- Tada, H., Paris, P.C., and Irwin, G.E. 1985 The Stress Analysis of Cracks Handbook , 2nd ed. Paris Productions, Inc., St. Louis, MO.
- Wöhler, A. Versuche über die Festigkeit der Eisenbahnwagenachsen, Zeitschrift für Bauwesen , Vol. 10, 1860; English summary (1867), Engineering , Vol. 4, 160-1.
- Neuber, H., Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law. J. of Appl. Mech. , ASME Transactions, Vol. 8, pp. 544–50, 1961.
- Fatemi, A. and Socie, D.F., A critical plane approach to multiaxial fatigue damage including out- of-phase loading, Fatigue Fract. Eng. Mater. Struct ., Vol. 11, No. 3, 1988, p. 149.
- Paris, P.C., Gomez, M.P., and Anderson, W.P., A rational analytic theory of fatigue, The Trend in Engineering , Vol. 13, pp. 9–14, 1961.
- Paris, P.C. and Erdogan, F., A critical analysis of crack propagation laws, J. of Basic Eng. , Vol. 85, pp. 528–34, 1963.
11.5 Design Optimization
Nam Ho Kim
Introduction
The design of a structural system has two categories: designing a new structure and improving the existing structure to perform better. The design engineer’s experience and creative ideas are required in the development of a new structure, since it is difficult to quantify a new design using mathematical measures. Recently, limited inroads have been made in the creative work of the structural design using mathematical tools.^1 However, the latter evolutionary process is encountered much more frequently in engineering designs. For example, how many times does an automotive company design a new car using a completely different concept? The majority of a design engineer’s work concentrates on improving the existing vehicle so that the new car can be more comfortable, more durable, and safer. In this section, we will focus on a design’s evolutionary process by using mathematical models and computational tools. Structural design is a procedure for improving or enhancing the performance of a structure by changing its parameters. A performance measure, which can be quite general in engineering fields, can include the following: the weight, stiffness, and compliance of a structure; the fatigue life of a mechanical component; the noise in the passenger compartment; the vibration level; the safety of a vehicle in a crash, and so forth. However, this does not address such aesthetic measures as whether a car or a structural design is
11 -46 Section 11
attractive to customers. All performance measures are presumed to be measurable quantities. System parameters are variables that a design engineer can change during the design process. For example, the thickness of a vehicle body panel can be changed to improve vehicle performance. The cross section of a beam can be changed in designing bridge structures. System parameters that can be changed during the design process are called design variables , even including the geometry of the structure. Great strides have been made during the past decade in computer-aided design (CAD) and computer- aided engineering (CAE) tools for mechanical system development. Discipline-oriented simulation capa- bilities in structures, mechanical system dynamics, aerodynamics, control systems, and numerous related fields are now being used to support a broad range of mechanical system design applications. Integration of these tools to create a robust simulation-based design capability, however, remains a challenge. Based on their extensive survey of the automotive industry in the mid-1980s, Clark and Fujimoto 2 concluded that simulation tools in support of vehicle development were on the horizon but not yet ready for pervasive application. The explosion in computer, software, and modeling and simulation technology that has occurred since the mid-1980s suggests that high-fidelity tools for simulation-based design are now at hand. Properly integrated, they can resolve uncertainties and significantly affect mechanical system design. Modern developments of structural design are closely related to concurrent engineering environments by which multidisciplinary simulation, design, and manufacturing are possible. Even though concurrent engineering is not the focus of this section, we want to emphasize structural design as a component of concurrent engineering. An important feature of the concurrent engineering is database management using the CAD tool. Structural modeling and most interfaces are achieved using the CAD tool. Thus, design parameterization and structural model updates have to be carried out within the CAD model. Through the design parameterization, CAD, CAE, and CAM procedures are interrelated to form a concurrent engineering environment. The structural engineering design in the simulation-based process consists of structural modeling, design parameterization, structural analysis, design problem definition, design sensitivity analysis, and design optimization. Figure 11.5.1 is a flowchart of the structural design process in which computational analysis and mathematical programming play essential roles in the design. The success of the system- level, simulation-based design process strongly depends on a consistent design parameterization, an accurate structural and design sensitivity analysis, and an efficient mathematical programming algorithm. A design engineer simplifies the physical engineering problem into a mathematical model that can represent the physical problem up to the desired level of accuracy. A mathematical model has parameters that are related to the system parameters of the physical problem. A design engineer identifies those design variables to be used during the design process. Design parameterization , which allows the design engineer to define the geometric properties for each design component of the structural system being designed, is one of the most important steps in the structural design process. The principal role of design parameterization is to define the geometric parameters that characterize the structural model and to collect a subset of the geometric parameters as design variables. Design parameterization forces engi- neering teams in design, analysis, and manufacturing to interact at an early design stage, and supports a unified design variable set to be used as the common ground to carry out all analysis, design, and manufacturing processes. Only proper design parameterization will yield a good optimum design, since the optimization algorithm will search within a design space that is defined for the design problem. The design space is defined by the type, number, and range of design variables. Depending on whether it is a concept or detailed design, selected design variables could be non-CAD based parameters. An example of such a design variable is a tire’s stiffness characteristic in vehicle dynamics during an early vehicle design stage. Structural analysis can be carried out using experiments in actual or reduced scale, which is a straight- forward and still prevalent method for industrial applications. However, the expense and inefficiency involved in fabricating prototypes make this approach difficult to apply. The analytical method may resolve these difficulties, since it approximates the structural problem as a mathematical model and solves
11 -48 Section 11
Cost and constraints can be defined by combining certain performance measures with appropriate constraint bounds for interactive design optimization. The cost function , sometimes called the objective function , is minimized (or maximized) during optimization. Selection of a proper cost function is an important decision in the design process. A valid cost function has to be influenced by the design variables of the problem; otherwise, it is not possible to reduce the cost by changing the design. In many situations, an obvious cost function can be identified. In other situations, the cost function is a combination of different structural performance measures. This is called a multiobjective cost function. Constraint functions are the criteria that the system has to satisfy for each feasible design. Among all design ranges, those that satisfy the constraint functions are candidates for the optimum design. For example, a design engineer may want to design a bridge whose weight is minimized and whose maximum stress is less than the yield stress. In this case, the cost function, or weight, is the most important criterion to be minimized. However, as long as stress, or constraint, is less than the yield stress, the stress level is not important. Design sensitivity analysis is used to compute the sensitivity of performance measures with respect to design variables. This is one of the most expensive and complicated procedures in the structural opti- mization process. Structural design sensitivity analysis is concerned with the relationship between design variables available to the engineer and the structural response determined by the laws of mechanics. Design sensitivity information provides a quantitative estimate of desirable design change, even if a systematic design optimization method is not used. Based on the design sensitivity results, a design engineer can decide on the direction and amount of design change needed to improve the performance measures. In addition, design sensitivity information can provide answers to “what if ” questions by predicting performance measure perturbations when the perturbations of design variables are provided. Substantial literature has emerged in the field of structural design sensitivity analysis. 4 Design sensitivity analysis of structural systems and machine components has emerged as a much-needed design tool, not only because of its role in optimization algorithms but also because design sensitivity information can be used in a computer - aided engineering environment for early product trade - off in a concurrent design process. Recently, the advent of powerful graphics - based engineering workstations with increasing computa- tional power has created an ideal environment for making interactive design optimization a viable alternative to more monolithic batch - based design optimization. This environment integrates design processes by letting the design engineer create a geometrical model, build a finite element model, parameterize the geometric model, perform FEA, visualize FEA results, characterize performance mea- sures, and carry out design sensitivity analysis and optimization. Design sensitivity information can be used during a postprocessing of the interactive design process. The principal objective of the postprocessing design stage is to utilize the design sensitivity information to improve the design. Figure 11.5.2 shows the four - step interactive design process: (1) to visually display design sensitivity information, (2) to carry out what - if studies, (3) to make trade - off determinations, and (4) to execute interactive design optimization. The first three design steps, which are interactive modes of the design process, help the design engineer improve the design by providing structural behavior information at the current design stage. The last design step, which could be either interactive or a batch mode of the design process, launches a mathematical programming algorithm to perform design opti- mization. Depending on the design problem, the design engineer could use some or all of the four design steps to improve the design at each iterative step. As a result, new designs could be obtained from what - if, trade - off, or interactive optimization design steps. For the purposes of design optimization, a mathematical programming technique is often used to find an optimum design that can best improve the cost function within a feasible region. Mathematical programming generates a set of design variables that require performance values from structural analysis and sensitivity information from design sensitivity analysis to find an optimum design. Thus, the struc- tural model has to be updated for a different set of design variables supplied by mathematical program- ming. If the cost function reaches a minimum with all constraint requirements satisfied, then an optimum design is obtained.
Engineering Design 11 -
Structural Design Parameterization
In structural modeling, the physical problem is represented by mathematical expressions, which contain parameters for defining that problem. For example, the cantilever beam in Figure 11.5.3 has parameters that include the length l , the radius of cross section r , and Young’s modulus E. These parameters, which define the system, are called design variables. If design variables are determined, then the structural problem can be analyzed. Obviously, different design variable values usually yield different analysis results. The aim of the structural design process is to find the values of design variables that satisfy all requirements. All design variables must satisfy the physical requirements of the problem. For example, length l of the cantilever beam in Figure 11.5.3 cannot have a negative value. Physical requirements define the design variable bounds. Valid design variables may have to take into account various manufacturing require- ments. For example, the radius of a cantilever beam satisfies its physical requirement if r is a positive number. However, in real applications, the circular cross - sectional beam may not be manufactured if its radius is bigger than r^0. Thus, the range of feasible design can be stated as 0 < r £ r^01. In addition, the design engineer may want to impose certain design constraints on the problem. For example, the maximum stress of the beam may not exceed s 0 and the maximum tip displacement of the beam must not be greater than z^0. A set of design variables that satisfy the constraints is called a feasible design , whereas a set that does not satisfy constraints is called an infeasible design. It is difficult to determine whether a current design is feasible unless the structural problem is analyzed. For complicated structural problems, it may not be simple to choose the appropriate design constraints so that the feasible region is not empty. There are two types of design variables: continuous and discrete. Many design optimization algorithms consider design variables to be continuous. In this section, we presume that all design variables are
FIGURE 11.5.2 Postprocessing design stage.
FIGURE 11.5.3 Parameters defining circular cross - sectional cantilever beam.
(^1) In general, the bounds of design variables are denoted as r L (^) £ r £ r U (^) where r L (^) is called the lower bound and r U
is called the upper bound, respectively.
Yes
Interactive Design Mode
Structural Analysis
Design Sensitivity Analysis
Sensitivity Display What-If study Trade-Off Design Optimization
Update Structural Model
Stop
Optimized? No
E r
F
l
Engineering Design 11 -
In this section, several possible design parameterizations, such as constant and linear designs as described in Figure 11.5.5, are introduced for the line and surface design components. These are not at all the only possible design parameters, and other more complicated design parameterizations can be used. However, the method presented in this section can be extended to other complicated design parameterizations. One important thing to consider when more complicated design parameterizations are used is that the finite element model must be sophisticated enough to support the design parame- terization method used. Geometric parameters can be defined at the end grid points of a line, or at the corner points of a surface. A bilinear thickness distribution can be used to characterize a surface design component, as shown in Figure 11.5.5(b). Note that each dimension that defines the cross-sectional shape, such as width and height in Figure 11.5.5(a), can be treated as a design variable, and be allowed to vary to the same degree as the corresponding variable at the other end (constant parameterization), or to a different degree (linear parameterization). Moreover, in order to maintain design continuity for a symmetric design, or to reduce the number of design variables, design variables can change either independently of, or proportionally to, certain variables across design components through design vari- able linking, as shown in Figure 11.5.6.
Line Design Components A three-dimensional line component can be used for truss or beam design components. A truss design component can handle tensile and compressive load and may be composed of several truss finite elements. A beam design component can handle tensile, compressive, and bending load. A linearly tapered cross- sectional shape can be considered within the design component. There are three cross-section types for this design component: symmetric, un-symmetric, and general. Figure 11.5.7 illustrates the geometry of a line design component.
FIGURE 11.5.5 Line and surface design parameterization.
FIGURE 11.5.6 Design variable linking.
Constant Parameterization Linear Parameterization Bilinear Parameterization (a) Line Design Component (b) Surface Design Component
u 2 u 3
u 1
u 4
u 1
u (^2)
u 3 u 4
u 1 =u u 2 =u
u 3
u
u 1
u u 1 ?u 3 u 2 ?u 4
u
u 3
u (^4)
u 2
u
u 3
u 4
u 2
Surface 1 Surface 2
u 3 of surface 1 =u2 of surface 2 u 4 of surface 1 =u1 of surface 2
11 -52 Section 11
Material properties, such as mass density and Young’s modulus, and geometric parameters that define the cross-sectional shape can be taken as design variables along the axial axis x 1. All material property design variables are assumed to be constant along the design component’s axial axis. The geometric design variables that can linearly vary along the axis of the design component are the dimensions of each cross-section, that is, r for a solid and hollow circular cross section, and b and h for other cross sections, as shown in Table 11.5.1 and Table 11.5.2.
FIGURE 11.5.7 Line design component.
TABLE 11.5.1 Symmetric Cross Sections
Cross-Sectional Shape
Design Variable a 1 2 3 4 5 6
Solid circular r E r — — —
Solid rectangular h b E r — —
Hollow circular r t E r — —
Hollow rectangular h b t w E r
I-section h b t w E r
a (^) E is Young’s modulus and r is the mass density.
x3 ,z
I
J
X 2
X 1
X 3
x1 ,z
x 2 ,z xp 2 xp
z 4
l r b h r t w
b
h
t
t
w
h
b
11 -54 Section 11
Surface Design Components A membrane component can handle both an in-plane tensile and a compressive load. A shell component can handle an in-plane tensile, compressive, and bending load. The design component may be composed of several membrane/shell finite elements. Figure 11.5.8 illustrates the geometry of a membrane/shell design component. Surface component design variables include thickness, mass density, and Young’s modulus. Surface thickness is parameterized using a bilinear shape function. Four geometric design variables are defined for each surface design component: thickness hI , hJ , hK , and hL at grid points I , J , K , and L , respectively, as shown in Figure 11.5.8 and Table 11.5.4. The four-node quadrilateral surface component can be reduced to a triangular surface component by defining duplicate node numbers for the third and fourth ( K and L ) node locations. If node L is not defined, then it defaults to node K. The design component thickness is assumed to vary bi-linearly inside the design component.
TABLE 11.5.3 General Cross Sections
General Cross-Sectional Shape
Design Variable 1 2 3 … 2 n +1 2 n +
s 1 t 1 s 1 … E r
FIGURE 11.5.8 Surface design component.
TABLE 11.5.4 Design Variables of Surface Design Component Design Variables 1 2 3 4 5 6 E r h (^) I h (^) J h (^) K h (^) L
s 1 ,t
s 2 ,t 2
s 3 ,t
sn ,tn
J
X 2 X 1
X 3
x2 x
x
I
K
L
hL h(x 1 ,x 2 ) hJ
hI
hK
I J
K,L
(Triangular option)
Ω
Engineering Design 11 -
The Shape Design Variable
While material property and the sizing design variables are related to the parameters of the structural problem, the shape design variable is related to the structure’s geometry. The shape of the structure does not explicitly appear as a parameter in the structural formulation. Although the design variables in Figure 11.5.4 determine the cross - sectional shape, they are not shape design variables, since these cross - sectional shapes are considered parameters in the structural problem. However, the length of the truss or beam should be treated as a shape design variable. Usually, the shape design variable defines the domain of integration in structural analysis. Thus, it is not possible to extract shape design variables from a structural model and to use them as sizing design variables. Consider a rectangular block with a slot, as presented in Figure 11.5.9. The location and size of the slot is determined by the geometric values of Cx , Cy , D (^) y , r 1 , and r 2 , which are shape design variables. Different values of shape design variables yield different structural shapes. However, these shape design variables do not explicitly appear in the structural problem. If the finite element method is used to perform structural analysis, then integration is carried out over the structural domain (the gray area), which is the shape design variable. Since shape design variables do not explicitly appear in the structural problem, the shape design problem is more difficult to solve than the sizing design problem. Shape design parameterization, which describes the boundary shape of a structure as a function of the design variables, is an essential step in the shape design process. Inappropriate parameterization can lead to unacceptable shapes.5,6^ To parameterize the structural boundaries and to achieve optimum shape design, boundary shape can be described in three ways: (1) by using boundary nodal coordinates, (2) by using polynomials, or (3) by using spline blending functions. However, it is important to point out that there are many methods of parameterization and that the methods presented in this section are only a few of them, including complicated parameterization methods developed in commercial CAD tools. One important aspect of shape design parameterization is the connection of the design parameterization to the computation of the design velocity field. In the first method, boundary nodal coordinates of the finite element model are used as shape design variables. Although the method is simple and easy to use, it has the following drawbacks: (1) the number of design variables tends to become very large, which may lead to high computational costs and optimi- zation problems that are difficult to solve; (2) the first derivative of the design boundary is not continuous across boundary nodes, which may lead to an unacceptable or impractical design; and (3) computational accuracy is not ensured, since it is difficult to maintain an adequate finite element mesh during the optimization process. One can use coordinates of selected master nodes as shape design variables and employ an isoparametric mapping to generate a finite element mesh. Several methods have been developed to parameterize the design boundary with polynomials. Bhavikatti and Ramakrishnan^7 used a 5th-degree polynomial, with the coefficients taken as design variables, to parameterize the boundary shape of a rotating disk. Prasad and Emerson 8 used a similar approach to optimize an engine connecting rod. In a more general approach, such as that used by Kristensen and Madsen^9 and Pedersen and Laursen,^10 the boundary is parameterized using a linear combination of shape
FIGURE 11.5.9 Shape design variables.
Cx
Cy
Dy Dy
r 1
r (^2)
Cx
Cy
r 1
r 2
Engineering Design 11 -
From Equation 11.5.3 and Equation 11.5.4, note that the slope of the cubic line can change its sign only twice, and that the curve can have only one inflection point. Consequently, PC entities such as PC lines and PC patches minimize the possibility of yielding oscillating boundaries during the design process.^6 However, geometric entities with predefined or sophisticated shapes, such as a circular hole, cannot be represented by a single cubic curve. To minimize modeling errors, such a boundary can be broken into small pieces. These pieces are then “glued” together in the design process as one geometric feature by appropriately linking design variables. For shape design, planar parametric cubic lines and spatial parametric bicubic patches are utilized to represent the design boundaries of two-dimensional and three-dimensional structural components, respectively. There are a number of methods for creating geometric entities, for example, defining four control points to create a Bezier curve, or constructing four edge curves to create a surface. Although geometric entities have different characteristics depending on the way they are created, they are nevertheless always represented by polynomials with the same order regardless of the way they are created. Consequently, a single geometric entity can be created using a variety of methods. For example, the planar curve shown in Figure 11.5.10(a) is created by defining four distinct points in the plane, and is thus called a four- point curve. The mathematical expression of the four-point curve is given as
The same curve shown in Figure 11.5.10(b) can also be created by giving the position and slope at the endpoints, referred to as geometric coefficients:
(a) (b) FIGURE 11.5.10 Planar parametric cubic curves (a) curve created by four points; (b) curve created by geometric coefficients.
z (^) , u ( ) u = 3 s u 3^2 + 2 s u 2 + s 1
z (^) , uu ( ) u = 6 s u 3 + 2 s 2
?
x u u
y u u u u
p
p
p
p
0
1
0
1
[ , ]
[ , ]
[ ,. ]
[ ,. ]
u
u
x
y
p 3 = p (1) = (3,2)
p 1 = p (1/3) = (1,1) p 2 = p (2/3) = (2,1)
p 0 = p (0) = (0,0) x
y
p 1
p u 0
p 0
p u 1
11 -58 Section 11
Therefore, one curve can be represented using different methods. However, these representations of curves and patches are mathematically equivalent, and one can be transformed into another by using certain linear transformations. For example, the geometric coefficients of a curve can be transformed into a four-point format, so the shape of the curve can be controlled according to the position of the four points. In fact, geometric coefficients are selected as unified geometric data, independent of the methods used to create geometric entities. Because design parameterization has the versatility of repre- senting the same geometric entity in different ways, it can be systematically developed to provide the design engineer with sufficient resources for solving a wide variety of shape design problems. The design parameterization method presented in this section is a three-step process, as illustrated in Figure 11.5.11. The first step is to create a geometric feature by grouping a number of interconnected geometric entities together, and by defining the type of geometric feature. The design engineer identifies the geometric entities that form the geometric features. The second step is to define the design variables within each geometric feature. Geometric features that are frequently used in the construction of structural components can be put in the library of predefined geometric features. The design engineer can then parameterize these predefined features simply by selecting associated predefined shape design variables. To demonstrate the use of the library, two predefined geometric features have been defined, a circular hole and a tapered slot, as shown in Figure 11.5.12. For the circular hole, which is formed by connecting a number of circular arcs end to end with the same center point and radius, both the radius and center point of the circle can be defined as shape design variables. For the tapered slot, which is formed by connecting a number of straight lines and circular arcs, length dp3, radii dp4 and dp5, and center point dp1 and dp2 can all be defined as shape design variables. The design parameterization flow for the predefined geometric feature is illus- trated in Figure 11.5.11(c).
(a)
(b)
FIGURE 11.5.11 Shape design parameterization of features (a) overall design parameterization process; (b) param- eterization for a user-defined feature; (c) parameterization for a predefined feature.
Form Geometry Features
Predefined Features?
Predefined Parameterization
User Defined Parameterization
No Yes
Design Parameter Linking
Parameterized Geometric Model
First Step
Second Step
Third Step
Define Geometry Entity Type
Assign Parameters for Each Geometry Entity
Link Parameters Across Geometry Entities
Parameterized User Defined Feature
Assign Parameters for The Geometry Feature
Parameterized Pre-defined Feature
11 -60 Section 11
After design variable linking, only one design variable, dp1, the beam length represented by the x -coordinate of grid #3 in line #102, is allowed to vary independently. The parameterization flow for the user-defined feature is shown in Figure 11.5.11(b). The shape design parameterization procedure for a user-defined geometric feature, as illustrated above, is summarized as follows:
- Identify the types of geometric entities that are to be used to construct the user-defined geometric feature.
- Parameterize the geometric entities by defining free and proportional design variables in each geometric entity.
- Generate the parameterized geometric feature by linking free design variables across geometric entities. Each predefined geometric feature that can be included in the feature library, such as the circular hole and tapered slot, is preconstructed by using this procedure. If necessary, the third step is designed to link design variables across geometric features. For example, a cantilever beam with a circular hole, as shown in Figure 11.5.14, is to be parameterized so that the position of the hole is proportional to the beam length. The x -coordinate of the circular hole Hh can be parameterized by using the predefined parameterization process, as described in Figure 11.5.12(a). The length of beam Hb can be parameterized as a user-defined feature, as illustrated in Figure 11.5.13. With the two parameterized features, the x -coordinate of the hole can be linked to the beam length. As described here, fundamental shape design parameterization is defined within geometric entities, and the parameterized features are created by using geometric entities. A hierarchy of the design parameterization method to build a parameterized geometric model is shown in Figure 11.5.15.
FIGURE 11.5.14 Design variable linking across parameterized geometric features.
FIGURE 11.5.15 Hierarchy of shape design parameterization.
Hh y Hb
x
Geometric Entity #
Geometric Entity #n
Geometric Entity #
Geometric Entity # •••
Parameterized Geometric Feature #
Parameterized Geometric Feature #
Parameterized Geometric Feature #N
•••
Parameterized Geometric Model
Parameterized Geometric Entities
Link Geometric Entities
Parameterized Geometric Features
Link Geometric Features
Parameterized Geometric Model
Engineering Design 11 -
Curve Design Parameterization For a two-dimensional shape design, the boundaries are planar curves. In general, there are eight degrees of freedom for a planar cubic curve, as expressed in Equation 11.5.1 and Equation 11.5.2, with z ( u ) serving as the constant. Planar curves with eight degrees of freedom are designated as basic curves, while predefined curves that are constrained, such as a circular arc, are designated as specialized curves. A specialized curve has fewer degrees of freedom since some of the basic degrees of freedom are linked (constrained) in order to define the required characteristics of the curve. From a computational point of view, algebraic and geometric curves are the most interesting among the six basic types of curves. All parametric cubic entities can be transformed into various other formats by using certain linear transformations. For shape design, three major transformations are necessary: (1) from the geometric coefficient matrix B to the design variable matrix G to compute design variable values; (2) from matrix G to matrix B to update geometric entities for a perturbed design shape; and (3) from matrix G to the algebraic coefficient matrix A to compute the boundary velocity field. For the basic curves, the transformation from matrix G to matrix B for each curve format can be described by their corresponding 4 • 4 constant matrices. The curve format transformations are summarized in Figure 11.5.16.
Surface Design Parameterization For three-dimensional shape design, design boundaries are surfaces in space. In general, there are 48 degrees of freedom for a parametric bicubic surface. For a parametric bicubic surface, the x -, y -, and z -components can be expressed using the three functions, as
(a)
(b)
(c)
FIGURE 11.5.16 Curve format transformations for two-dimensional shape design (a) transformations from B to G ; (b) transformations from G to B ; (c) transformations from B to A.
Geometric Coefficients B
Bezier Curve G = Q −^1 MB
B-Spline Curve G = R −^1 MB
Spline Curve G = V −^1 MB
Four-Point Curve G = K −^1 B
Geometric Coefficients B
Four-Point Curve G
B-Spline Curve G
B = KG
Bezier Curve B^ =^ M −^1^ QG G Spline Curve G
B = M −^1 VG
B = M −^1 RG
Geometric Coefficients B
Algebraic Coefficients A
A = MB
(Compute Boundary Velocity)
x x u w
y y u w
z z u w
Engineering Design 11 -
their orientation in space, the transformation between the local and global coordinates also changes. Thus, this transformation can be considered the configuration design variable. Since configuration design variables are defined for built-up structures, they are inherently coupled with shape design variables. That is, in order to allow one member of the built-up structure to rotate, another member’s shape needs to be changed. The configuration design variable is not applicable to solid components in which all rotations can be expressed in terms of shape changes.
The Topology Design Variable
If shape and configuration design variables represent changes in structural geometry and orientation, then topology design determines the structure’s layout. For example, in Figure 11.5.9, shape design can change the size and location of the slot within the block. However, shape design cannot completely remove the slot from the block, or introduce a new slot. Topology design determines whether the slot can be removed or an additional slot is required. The choice of the topology design variable is nontrivial compared to other design variables. Which parameter is capable of representing the birth or death of the structural layout? Early developments in topology design focused on truss structures. For a given set of points in space, design engineers tried to connect these points using truss structures, in order to find the best layout to support the largest load. Thus, the on - off types of topology design variables are used. These kinds of designs, however, could turn out to be discontinuous and unstable. Recent developments in topology design are strongly related to FEA. The candidate design domain is modeled using finite elements, and then the material property of each element is controlled. If it is necessary to remove a certain region, then the material property value (e.g., Young’s modulus) will approach zero, such that there will be no structural contribution from the removed region. Thus, material property design variables could be used for the purpose of topology design variables. The on - off type of design variable can be approximated by using continuous polynomials in order to remove the difficulties associated with discrete design variables. In many applications, topology design is used at the concept design stage such that the layout of the structure is determined. After the layout is determined, sizing and shape designs are used to determine the detailed geometry of the structure. A final comment on design parameterization: it is desirable to have a linearly independent set of design variables. If one does not, then relations between design variables must be imposed as constraints, which may make the design optimization process expensive, as the number of design variables and constraints increase. Furthermore, if design variable constraints are not properly established, meaningless design results will be obtained after an extensive amount of computational effort. As mentioned before, this problem is strongly related to structural modeling, since a well - defined structural model should have an independent set of parameters to define the entire system. Even if defining a good model is not an easy task for a complicated design problem, the design engineer nevertheless has to define a proper and independent set of parameters as much as possible in the structural modeling stage.
Design Variable Linking across Geometric Entities
There are three methods for categorizing the shape design problem: (1) identifying the arbitrary boundary shape, (2) determining the dimensions of the predefined shape, and (3) finding the locations of the boundaries. A design variable linking process is discussed in this section in order to support and ensure continuity between boundaries, as well as to retain predefined boundary shapes. For a geometric feature that is formed by a set of B-spline curves, C^2 -continuity across curves is naturally retained. As described in the previous sections, for the first type of shape design boundary any basic curve or surface can be utilized for parameterization. If constrained boundary shapes are required for two- dimensional structures, such as a straight edge or circular arc, then that specific type of curve must be used. For three-dimensional structures, the constrained boundaries can be predefined, such as a plane, cylinder, ball, or surface of revolution. If the geometric feature is composed of more than one curve or surface, continuity across curves or surfaces must be retained by linking shape design variables at the joint
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point or curve. As mentioned before, C^0 -, C^1 -, and C^2 -continuities can be retained for two-dimensional structures by using the appropriate curve type and design variable linking. For three-dimensional prob- lems, C^2 -continuity is difficult to maintain. The following three examples illustrate how the design variable linking process can be used to support the three types of shape design problems.
Type One Problems — Identifying the Arbitrary Boundary Shape A two-dimensional beam with arbitrarily shaped holes, shown in Figure 11.5.18(a), illustrates a design variable linking process that supports the first type of shape design problem. Figure 11.5.18(b) shows that hole A is formed by three curves. To parameterize this hole, the curves are defined as geometric curves. The endpoints and tangent vectors at the endpoints of these connected curves are linked to retain C^0 - and C^1 -continuities, respectively. The shape of the hole can be changed due to endpoint movement, length, or the direction of the tangent vector, as shown in Figure 11.5.19. Hole shape change due to movement ddp 1 of grid 3 in the negative y -direction is shown in Figure 11.5.19(a), such that the tangent vectors of lines 1 and 3 at grid 3 are kept the same. Moreover, hole shape change due to scaling the tangent vector of line 3 at grid 3 by ddp 2 , shown in Figure 11.5.19(b), is such that the location of grid 3 and the direction of the tangent vector are kept the same. Finally, hole shape change due to change ddp 3 in the tangent vector direction of lines 1 and 3 is shown in Figure 11.5.19(c). To avoid meaningless designs, for example, a hole that penetrates the boundary edges of a beam, the geometric boundaries can be displayed after the design change. Furthermore, numerical limits can be defined for design variables in order to restrict the degree of design perturbation.
Type Two Problems — Determining the Dimensions of a Predefined Shape For the second type of shape design problem, specialized curves and surfaces are used to parameterize geometric entities. In addition, design variable linking may be carried out in the design process to group curves and surfaces that define the geometric feature. The design variable linking method for parameterizing the second type of shape design boundary can be described using an elliptic hole formed by four conic curves, as shown in Figure 11.5.20. Each conic curve has three points to control its shape; however, relative altitude r is kept constant. To vary the major axis, the x -coordinate of point p 7 is defined as the independent design variable dp1. Points p 6 and p 8 are linked to dp1, with a proportionality of 1.0. In addition, the x -coordinates of points 1, 2, and 3 are linked to dp1 with a proportionality of –1.0. The elliptic hole varies in shape when the major axis is perturbed by ddp1, as shown in Figure 11.5.21(a). Similarly, for a design change in the minor axis, the y -coordinate of point p 4 is defined as the independent design variable dp2. Points p 1 and
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FIGURE 11.5.18 Parameterization of arbitrarily shaped holes (a) beam with arbitrarily shaped holes; (b) Hole A formed by three curves.
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