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To perform optimization via variation of the boundaries of the holes, we represent these boundaries by b-splines, and introduce a number of master nodes in order to give the system sufficiently many design parameters for the optimization. We shall require symmetry about the horizontal mid-axis of the geometry and utilize link facilities implemented in CAOS to link the movements of master nodes above this line to the corresponding master nodes below. The design model is illustrated in Fig. 14. Pay lq 7 7 ee | : _y .. oe N. Olhoff et al., On CAD-integrated structural topology and design optimization

Figure 14 To perform optimization via variation of the boundaries of the holes, we represent these boundaries by b-splines, and introduce a number of master nodes in order to give the system sufficiently many design parameters for the optimization. We shall require symmetry about the horizontal mid-axis of the geometry and utilize link facilities implemented in CAOS to link the movements of master nodes above this line to the corresponding master nodes below. The design model is illustrated in Fig. 14. Pay lq 7 7 ee | : _y .. oe N. Olhoff et al., On CAD-integrated structural topology and design optimization

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Fig. 2. (a) A periodic microstructure with square holes rotated the angle 8. (b) A square cell with a square hole. Layered materials (used for topology generation in the example in Section 6.1) is just one possible choice of microstructure that can be applied. The important feature is to choose a microstructure that allows density of material to cover the complete range of values from zero (void) to one (solid), and that this microstructure is periodic so that effective properties can be computed (numerically) through homogenization (theory of cells). This excludes circular holes in square cells, while square or rectangular holes in square cells, see Fig. 2, are suitable choices of simple microstructures. For the case of a rectangular hole in a square cell, the
Fig. 2. (a) A periodic microstructure with square holes rotated the angle 8. (b) A square cell with a square hole. Layered materials (used for topology generation in the example in Section 6.1) is just one possible choice of microstructure that can be applied. The important feature is to choose a microstructure that allows density of material to cover the complete range of values from zero (void) to one (solid), and that this microstructure is periodic so that effective properties can be computed (numerically) through homogenization (theory of cells). This excludes circular holes in square cells, while square or rectangular holes in square cells, see Fig. 2, are suitable choices of simple microstructures. For the case of a rectangular hole in a square cell, the
Fig. 1. Construction of a layering of second rank.
Fig. 1. Construction of a layering of second rank.
Here, A,, /;, p; and E, denote cross-sectional area, length, specific weight and Young’s modulus for the i-th bar of the truss. The design variables of the problem, see (23), are A, and x,, where the latter symbol represents an element of the total set of variable components of position vectors of joints in the structure. Equations (26a, b) are side constraints for the design variables. For simplicity in notation, the remaining constraints are written for a single loading case. Equations (25) are constraints that may be specified for any displacement component for the joints of the truss. Our structural optimization system possesses capabilities for optimization 2-D and 3-D trus: structures under multiple loading conditions, using cross-sectional areas of bars and position: of joints as design variables. The system is called SCOTS (Sizing and Configuration Optimiza. tion of Truss Structures). The development is inspired by [42-44]. In the current setting weight minimization is the design objective, and constraints include stresses, displacement: and elastic as well as plastic buckling of bars in compression. The mathamoaticral nrnacrammina farmiulatinn ic
Here, A,, /;, p; and E, denote cross-sectional area, length, specific weight and Young’s modulus for the i-th bar of the truss. The design variables of the problem, see (23), are A, and x,, where the latter symbol represents an element of the total set of variable components of position vectors of joints in the structure. Equations (26a, b) are side constraints for the design variables. For simplicity in notation, the remaining constraints are written for a single loading case. Equations (25) are constraints that may be specified for any displacement component for the joints of the truss. Our structural optimization system possesses capabilities for optimization 2-D and 3-D trus: structures under multiple loading conditions, using cross-sectional areas of bars and position: of joints as design variables. The system is called SCOTS (Sizing and Configuration Optimiza. tion of Truss Structures). The development is inspired by [42-44]. In the current setting weight minimization is the design objective, and constraints include stresses, displacement: and elastic as well as plastic buckling of bars in compression. The mathamoaticral nrnacrammina farmiulatinn ic
Fig. 5. Truss interpretation of result in Fig. 4 with bar areas determined by sizing for fixed positions of joints. Fig. 4. Solution of topology optimization problem.
Fig. 5. Truss interpretation of result in Fig. 4 with bar areas determined by sizing for fixed positions of joints. Fig. 4. Solution of topology optimization problem.
N. Olhoff et al., On CAD-integrated structural topology and design optimization
N. Olhoff et al., On CAD-integrated structural topology and design optimization
Fig. 6. The result when both bar cross-sectional areas and positions of unrestrained joints are used as design variables.
Fig. 6. The result when both bar cross-sectional areas and positions of unrestrained joints are used as design variables.
Fig. 7. Initial geometry for pedestal bearing. This geometry completely fills the available space. N. Olhoff et al., On CAD-integrated structural topology and design optimization
Fig. 7. Initial geometry for pedestal bearing. This geometry completely fills the available space. N. Olhoff et al., On CAD-integrated structural topology and design optimization
Fig. 8. Optimized topology illustrated by filling the elements by lumps of material corresponding to their fina density.
Fig. 8. Optimized topology illustrated by filling the elements by lumps of material corresponding to their fina density.
Fig. 10. Shape optimized finite element model. Fig. 9. Shape design model of optimized topology. N. Olhoff et al., On CAD-integrated structural topology and design optimization the analysis model presumes that the joint, regardless of its type, provides full contact with the underlying surface in all cases.
Fig. 10. Shape optimized finite element model. Fig. 9. Shape design model of optimized topology. N. Olhoff et al., On CAD-integrated structural topology and design optimization the analysis model presumes that the joint, regardless of its type, provides full contact with the underlying surface in all cases.
Fig. 11. The final geometry could look something like this. The optimized geometry can be transferred back into the CAD system where the final geometrical adjustments are easily performed by the designer, yielding for instance the
Fig. 11. The final geometry could look something like this. The optimized geometry can be transferred back into the CAD system where the final geometrical adjustments are easily performed by the designer, yielding for instance the
Fig. 12. Initial geometry with loads and boundary conditions.
Fig. 12. Initial geometry with loads and boundary conditions.
Fig. 14. Design model with b-splines as hole boundaries. i.e., with this model, we manage to create a feasible design and save 5.2% of the volume.
Fig. 14. Design model with b-splines as hole boundaries. i.e., with this model, we manage to create a feasible design and save 5.2% of the volume.
N. Olhoff et al., On CAD-integrated structural topology and design optimization The fact that the possible volume reduction even with a b-spline model is rather modest leads to the suspicion that the three-hole topology is not well suited for a structure of this type. It is therefore tempting to start the redesign procedure by a topology optimization. As discussed in Section 2, the topology optimization requires a volume constraint to be defined. The topology optimization system will then distribute the available volume in the available domain such that the stiffness is maximized. The system enables the user to specify regions or boundaries which are required to be solid, that is, of density 1. We shall use this facility in the present example because the function of the structure requires that the outer contour, except for the left vertical symmetry boundary, remains unchanged. The original geometry with three circular holes of radius 150mm has a volume of ne « a eo ee ee | ee Oe ee Se ee ee eee
N. Olhoff et al., On CAD-integrated structural topology and design optimization The fact that the possible volume reduction even with a b-spline model is rather modest leads to the suspicion that the three-hole topology is not well suited for a structure of this type. It is therefore tempting to start the redesign procedure by a topology optimization. As discussed in Section 2, the topology optimization requires a volume constraint to be defined. The topology optimization system will then distribute the available volume in the available domain such that the stiffness is maximized. The system enables the user to specify regions or boundaries which are required to be solid, that is, of density 1. We shall use this facility in the present example because the function of the structure requires that the outer contour, except for the left vertical symmetry boundary, remains unchanged. The original geometry with three circular holes of radius 150mm has a volume of ne « a eo ee ee | ee Oe ee Se ee ee eee
The upper right corner of the frame has been removed. This part of the geometry has a function, but it is structurally insignificant, and can therefore be excluded from the shape optimization and added to the modified structure afterwards. This simplification greatly facilitates the generation of the design model. Figure 16 illustrates the modifications that have been imposed on the optimized topology and the resulting initial finite element model is shown in Fig. 17. This structure has the data
The upper right corner of the frame has been removed. This part of the geometry has a function, but it is structurally insignificant, and can therefore be excluded from the shape optimization and added to the modified structure afterwards. This simplification greatly facilitates the generation of the design model. Figure 16 illustrates the modifications that have been imposed on the optimized topology and the resulting initial finite element model is shown in Fig. 17. This structure has the data
Fig. 17. Initial finite element mesh of optimized topology.
Fig. 17. Initial finite element mesh of optimized topology.
Fig. 18. Final finite element model. The volume is reduced by 42% in comparison with the initial design with circular holes. The final geometry is a frame-like structure. The stress constraint is not active because, for practical reasons, a minimum thickness is specified for the members of the resulting geometry. Unfortunately, the geometry.of the solution introduces the problem of stability which is not covered by CAOS. It is also a problem that the generation of a suitable finite element mesh
Fig. 18. Final finite element model. The volume is reduced by 42% in comparison with the initial design with circular holes. The final geometry is a frame-like structure. The stress constraint is not active because, for practical reasons, a minimum thickness is specified for the members of the resulting geometry. Unfortunately, the geometry.of the solution introduces the problem of stability which is not covered by CAOS. It is also a problem that the generation of a suitable finite element mesh
Fig. 19. Example of final geometry slightly modified by the designer. The upper right corner has been added again.
Fig. 19. Example of final geometry slightly modified by the designer. The upper right corner has been added again.