17th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, 13-17 June, 2016
Thermal Topology Optimization in OptiStruct Software Xueyong Qu, Narayanan Pagaldipti, Raphael Fleury, Junji Saiki Altair Engineering, Inc. 38 Executive Park, Suite 200, Irvine, CA 92614 E-mail:
[email protected] Abstract: This paper presents an overview of the recently implemented thermal topology optimization capabilities in Altair OptiStruct software. Design sensitivity analysis and optimization have been enhanced to support thermal compliance responses in addition to existing temperature responses. Topology optimization can be performed for pure thermal optimization problems or coupled thermal-structural optimization problems. Numerical examples are presented to demonstrate the advantage of thermal compliance formulation over temperature response formulation. Keywords: topology optimization, thermal optimization, thermal compliance, sensitivity analysis
1. Introduction Thermal analysis is used to determine the temperature field and heat fluxes in a structure. Coupled thermal-structure analysis uses the temperature results obtained from thermal analysis as temperature load to perform structure analyses such as static stress analysis. It can be one-way coupling with the results of the thermal analysis used as loading in structural analysis but not vice versa, or two way coupling when structural analysis such as contact affects thermal analysis. Thermal and coupled thermal-structural analyses are widely used in engineering practices such as aerospace vehicle design, electronics cooling system design, and automotive powertrain design. Extensive literature has been published both on theory and application of structural optimization[1-6]. In multidisciplinary optimization thermal analysis can be coupled with other disciplines, particularly aerodynamic analysis. Rajadas et al.[7] performed an integrated aerodynamic-thermal shape optimization for turbine blades where both the external blade shape and the internal coolant hole shapes were optimized for improved aerodynamic and thermal criteria. Dulikravitch et al.[8] performed an aero-thermal optimization for internally cooled turbine blades. Gersborg-Hansen et al.[9] performed topology optimization of heat conduction problems using the finite volume method. Ryu et al.[10] studied thermal topology optimization using density dependent shape functions. Cho et al.[11] performed thermal topology optimization using the level set method. Li et al[12] studied topology optimization of thermally actuated compliant mechanisms. Izui et al[13] performed topology optimization of thermal deformation control structure using level set method. Deaton and Grandhi[14] studied stress-based topology optimization of thermal structures. Alexandersen et al[15] performed convection topology optimization using density method. Coffin and Maute[16] studied steady state and transient natural convection problem using the level set method. OptiStruct[17] is a general purpose finite element solver and design optimization software. It can solve linear and non-linear problems in applied mechanics such as structure, heat transfer, fluid-structure interaction, and mechanical systems. Qu et al[18] demonstrated sizing and shape thermal optimization with temperature response and coupled thermal structural optimization in OptiStruct[17]. The current paper presents recently implemented thermal topology capabilities in OptiStruct[17]. Conduction and convection thermal analysis are solved by the finite element method. Design sensitivity analysis and optimization have been enhanced to support thermal compliance responses in addition to previously implemented temperature responses. Thermal responses such as thermal compliance and temperatures responses are supported either as objective functions or constraints. They can be used simultaneously with responses from static analysis, normal modes analysis, buckling analysis, dynamic analysis and user defined responses. Numerical examples are provided to demonstrate the newly added capabilities and advantages of thermal compliance topology optimization.
2. Thermal analysis Conductive and convective thermal problems can be solved in OptiStruct[17]. The governing equation for linear steady state thermal analysis can be discretized by the finite element method, and expressed in the finite element form as [19] (1) ([K c ] + [H ]){T} = {p B } + {p H }+ pQ
{ }
1
where
[K c ] = ∫ [B] [k ][B]dv T
V
[H] = ∫ [N] [N]hds T
S
{p B } = ∫ [N]
T
f B ds
S
{p H } = ∫ [N] hT fl ds T
S
{p } = ∫ [N] Qdv T
Q
V
[Kc] is the conductivity matrix, [H] is the convection matrix, {T} is the unknown temperature, [N] is the shape function, [B] is the derivative of [N] with respect to coordinates, [k] is the material thermal conductivity coefficient matrix, h is the convective heat transfer coefficient, fB is the boundary heat flux, Tfl is the ambient fluid temperature, Q is the volumetric heat generation, S is the boundary surface that has heat exchange, and V is the volume. The system of linear equations is solved to find the nodal temperature vector {T}. A comprehensive set of scalar, 1-D, 2-D and 3-D elements for conductive and convective thermal analysis is implemented in OptiStruct[17]. Both isotropic and anisotropic thermal materials can be used. Composite layups is also available for thermal analysis. Nonlinear steady state analysis, thermal contact and transient thermal analysis are analysis-only features in OptiStruct[17]. 3. Coupled thermal-structural analysis 3.1 Coupled Thermal-Structural Sensitivity Analysis In coupled thermal structural analysis, thermal analysis is performed first to determine the temperature field of the structure. The temperature field is used as temperature load for subsequent structural analysis. A single finite element mesh is usually used for both thermal and structural analyses. The finite element governing equation for static structural analysis is [17] K {D} = {f }+ {f T } (2) where [K] is the global stiffness matrix, {D} is the displacement vector, {fT} is the temperature loading, and {f} represents structural loading such as forces, pressures, etc.
[ ]
The coupling in thermal-structural analysis discussed in this paper is sequential. Thermal analysis affects subsequent structural analysis but structural analysis usually has no influence over thermal analysis. However, in coupled thermal-structural optimization, the optimizer modifies structural design parameters to satisfy constraints with improved objectives, which in turn affects thermal analysis. In OptiStruct[17], coupled thermal-structural analysis is performed in one single analysis run. Thermal analysis subcase and coupled structural analysis subcase are put together in the same input deck, where structural subcase use TEMP case control card to point to thermal analysis. Thermal analysis subcase is solved first to obtain temperature results, which is used as temperature load in subsequent coupled thermal-structural analysis. 3.2 Coupled Thermal-Structural Sensitivity Analysis In coupled thermal-structural optimization problems, sensitivities for structural responses that are influenced by thermal loading need to account for the sensitivity of the temperature loading. It has been shown by Qu et al[16] that this update is critical for thermal driven coupled thermal-structural optimization problems.
D K
−1
fT
K
dD = dxi
f
The equations are shown as follows, differentiating Eq. 2 by xi yields the sensitivities of structural displacements.
d d d [ + − dxi dxi dxi
]
In Eq. 3, the sensitivities of the thermal loading {fT} can be further expanded as follows.
2
(3)
T
f TT
fT
fT
d ∂ ∂ = + dxi ∂xi ∂
d dxi
(4)
The sensitivity analysis in OptiStruct[17] has both Eq. 3 and Eq. 4.
4. Optimization formulation and schemes The optimization problem can be formulated as Minimize f ( X )
Subject to g j ( X ) − g j ≤ 0, U
L
U
xi ≤ xi ≤ xi , where f ( X ) represents the objective function, g j ( X ) and
j = 1,..., M
(5)
i = 1,..., N U gj
represent the j-th constraint response and its upper L
U
bound, respectively. M is the total number of constraints; xi is the i-th design variable, xi and xi represent its lower and upper bounds, respectively. The total number of design variables is N. In Optistruct[17], design variables include: (1) sizing variables that define the cross-sectional dimensions of 1-D elements (rods and beams) and 2-D elements (plates and shells); (2) shape variables that define the shape variation of existing boundaries; and (3) topology variables that define the generalized material distribution allowing topological changes to the structure. The objective function and design constraints can be any of the responses supported in OptiStruct including volumes or weights of structural parts, structural compliance, eigenfrequencies, displacements, stresses, temperatures, and user defined responses through equation or external subroutines. Figure 1 shows the optimization scheme in OptiStruct software[17]. Advanced approximation techniques that use intermediate variables and intermediate responses are employed to reduce computational cost. Direct and adjoint methods are used to perform design sensitivity analysis. Several optimizers such as SQP are available to search for the optimum design. The details of OptiStruct optimization are published by Zhou et al.[6]
Figure 1. Optimization scheme in the OptiStruct software
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5. Thermal Compliance Response and Sensitivity Analysis In this paper, thermal compliance is implemented as a response in OptiStruct[17]. Thermal compliance can be used as objective function and/or constraint. It can be used together with all existing responses such as volume, displacement, etc. 5.1 Thermal compliance Thermal compliance[5] is defined as
TCOMP = 1 / 2 × T T P = 1 / 2 × T T [K c + H]T
(6)
where [Kc] is the conductivity matrix, [H] is the convection matrix, {T} is the unknown temperature. When thermal compliance is minimized, temperature at grids where power is applied is minimized, which typically is highest in the structure. Since it is a smooth convex function, optimization converges much quicker than minimizing the maximum temperature of the entire structure. 5.2 Thermal compliance sensitivity analysis
P
∂(K c + H) d (TCOMP ) ∂ 1 = TT − TT T dxi ∂dxi ∂xi 2
(7)
Once temperature vector is solved by finite element analysis, thermal compliance sensitivity can be calculated directly from Eq. 7. Topology, sizing and shape sensitivity analysis is implemented for thermal compliance response. 5.3 Thermal compliance optimization formulation When power is applied to the structure, thermal compliance should be minimized in order to obtain maximum conduction as shown by Eq. 8. This formulation results in lowest temperature
TCOMP = 1 / 2 × T T P = 1 / 2 × P T [K c + H ]−1 P
(8)
When enforced temperature is applied to the structure, temperature field is fixed. Thermal compliance should be maximized in order to obtain maximum conduction as shown by Eq. 9.
TCOMP = 1 / 2 × T T P = 1 / 2 × T T [K c + H]T
(9)
Convection is very common in heat transfer problem. Both enforced temperature and applied power are applied to the structure when convection exists. Optimization formulations for convection will be explored in the future and are not in the scope of the present work.
6. Numerical examples Three numerical examples are provided to demonstrate the capabilities implemented in OptiStruct[17]. 6.1 Classic thermal compliance optimization A flat plate structure is subject to flux input to the entire surface and fixed temperature at the top middle edge, shown in Figure 2. Optimization problem is shown by Eq. 10. Minimize TCOMP Such that volfrac ≤ 0.3
(10)
where TCOMP is the thermal compliance of the structure and volfrac is the volume fraction of the structure. Gersborg-Hansen et al[9] solved this problem and obtained a tree type of structure. OptiStruct[17] optimization converges in 56 iterations. The optimal design is a tree like structure and is shown in Figure 3.
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Figure 2. Plate model subjected to flux input to the entire surface and fixed temperatures at top middle edge
Figure 3. Final optimum design 6.2 2D thermal compliance optimization A 2D plate structure is subject to flux input to the top middle edge and fixed temperature at bottom two corners, shown in Figure 4. Optimization problem is shown by Eq. 11. Minimize TCOMP Such that volfrac ≤ 0.3
5
(11)
where TCOMP is the thermal compliance of the structure and volfrac is the volume fraction of the structure. This optimization converges in 25 iterations. The optimal design is a truss like structure to provide maximum conduction as shown in Figure 5.
Figure 4. Plate model subjected to flux input to the top middle edge and fixed temperature at bottom corners
Figure 5. Final optimal design
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6.3 Advantage of thermal compliance optimization A 3D solid beam structure is subjected to flux input to the bottom surface as shown by Fig 6. The beam is consist of 126,000 hexa elements. Both ends and top surface of the beam have fixed temperatures. The remaining surfaces have adiabatic boundary conditions.
Figure 6. Solid beam model subjected to heat flux input and fixed temperatures The top and bottom of the beam have a layer of solid element shown in blue color, which are not in the design domain. Rest of the elements are in the topology design domain shown in green color. Total number of design element is 113,400. Two optimization studies are performed. One minimizes thermal compliance, the other minimizes the maximum temperatures of the entire structure. Minimizing thermal compliance also minimizes temperature at grids where power is applied. Those temperatures are typically the highest in the structure, therefore final designs of the two studies are similar. Figure 7. shows thermal compliance design on the left and minmax temperature design on the right. The design concepts are comparable.
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Figure 7. Thermal compliance design on the left, minmax temperature design on the right But the computational cost of minimizing thermal compliance is 34 times lower, shown in Table 1. There are two main reasons for the drastically reduced computational cost. Firstly, the sensitivity of thermal compliance is a single response, and does not require matrix inversion and forward-backward substitution unlike the sensitivity calculations of temperature responses. Furthermore, while thermal compliance is a single global response, the minmax temperature formulation needed to retains 500 critical temperature responses in order to achieve convergence, for which 500 response sensitivities need to be calculated. Secondly, since thermal compliance is a smooth convex function, the optimization converges much quicker than minimizing the maximum temperature of the entire structure. Table 1. Comparison of optimization solution time Minimizing thermal compliance Minimizing the maximum temperatures of the entire structure Total optimization solution time 2,113.7s 72,611.5s Total number of iterations 43 95
7. Conclusions and Future Work Thermal topology optimization capabilities have been added to OptiStruct. Thermal compliance response is supported in additional to existing temperature responses. A few optimization examples have been provided to demonstrate these capabilities. Thermal compliance optimization is shown to be significantly faster than previously implemented formulation of minimizing the maximum temperature of the entire structure. Coupled thermal structural optimization in OptiStruct currently uses the direct method and works well for shape and sizing optimization[18]. However, coupled thermal structural topology optimization requires adjoint thermal structural sensitivity analysis without which the computational costs are prohibitive. The adjoint formulation for the coupled problem will be implemented and compared against the direct method in the future. Optimization formulations with convection will also be explored.
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