Dear Teng,
Thank you very much for your kind words and comments. The following are some of my thoughts:
(1) I think the main contribution of this work is to provide a way to do topology optimization in an explicit and geometrical way, which cannot
be achieved easily in the current toology optimization framework. As mentioned in the manuscript, although superellipse curve is used
in the present work to represent the shape of a component explicitly, in fact, well-established NURBS and iso-parameteric modeling
techniques can also be employed to descibe the geometry of the component. In this sense, our approach establish a link between the
conventional shape optimization and topology optimization, which is currently under intensive investigation. Within the proposed
computational framework, in principle all mehthods developed for shape optimization can be employed topology optimization problems.
Futhermore, since our approach is geometrically explicit in nature, it provides a natural application yard for modern FEM analysis techniques,
such as Isogeometry Analysis. We intend to report the corresponding results in subsequent work.
(2) As for the issue of local minimum, first I would like to point out that since the objective functionals in topology optimization are usually not
lower-semi-continuous with respoct to the weak* topology of L^(infinity) sapce, usually regularization treatments are required to guarantee the
well-posedness. Under this circumstance, the topology of a optimal design is always not too complicated. As long as the final topology does
can be represeneted by the initial set of components, our approach can guarantee to include the global optimum in the design space and
therefore has the opportunity to find it. Since topology optimization problem is non-convex in nature, usually global optimum cannnot be
expected for both conventional and the proposed numerical solution approaches. As shown in our paper, standard benchmark examples,
our appoach can obtain the same results as those obtianed by conventional methods. Futhermore, since our optimization problem
is a finite dimensional one, convergence to local optimum can be guaranteed if appropriate optimization algorithms (e.g., SQP, SNOPT) are
adopted.
(3) You are right the optimum design for structural stiffness should use all available materials. Actually, in all our test examples, the volume
contsraints are active for optimal sloutions. The total material volume will not be reduced through “hiding” or "overlapping" of components
since the volume of "useful" components will expand smartly during the course of optimization.
(4) In the proposed framework, the overlapping region between two components can be easily identified by an "AND" operation on their respective
level set functions (i.e., x in the overlapping region iff min (\Phi_1(x), Phi_2(x)). Once the overlapping region is identified, we can construct
a inscribed circle of this region to estimate its "thickness".
Best regards，
Xu Guo
Dear Teng,
Thank you very much for your kind words and comments. The following are some of my thoughts:
(1) I think the main contribution of this work is to provide a way to do topology optimization in an explicit and geometrical way, which cannot
be achieved easily in the current toology optimization framework. As mentioned in the manuscript, although superellipse curve is used
in the present work to represent the shape of a component explicitly, in fact, well-established NURBS and iso-parameteric modeling
techniques can also be employed to descibe the geometry of the component. In this sense, our approach establish a link between the
conventional shape optimization and topology optimization, which is currently under intensive investigation. Within the proposed
computational framework, in principle all mehthods developed for shape optimization can be employed topology optimization problems.
Futhermore, since our approach is geometrically explicit in nature, it provides a natural application yard for modern FEM analysis techniques,
such as Isogeometry Analysis. We intend to report the corresponding results in subsequent work.
(2) As for the issue of local minimum, first I would like to point out that since the objective functionals in topology optimization are usually not
lower-semi-continuous with respoct to the weak* topology of L^(infinity) sapce, usually regularization treatments are required to guarantee the
well-posedness. Under this circumstance, the topology of a optimal design is always not too complicated. As long as the final topology does
can be represeneted by the initial set of components, our approach can guarantee to include the global optimum in the design space and
therefore has the opportunity to find it. Since topology optimization problem is non-convex in nature, usually global optimum cannnot be
expected for both conventional and the proposed numerical solution approaches. As shown in our paper, standard benchmark examples,
our appoach can obtain the same results as those obtianed by conventional methods. Futhermore, since our optimization problem
is a finite dimensional one, convergence to local optimum can be guaranteed if appropriate optimization algorithms (e.g., SQP, SNOPT) are
adopted.
(3) You are right the optimum design for structural stiffness should use all available materials. Actually, in all our test examples, the volume
contsraints are active for optimal sloutions. The total material volume will not be reduced through “hiding” or "overlapping" of components
since the volume of "useful" components will expand smartly during the course of optimization.
(4) In the proposed framework, the overlapping region between two components can be easily identified by an "AND" operation on their respective
level set functions (i.e., x in the overlapping region iff min (\Phi_1(x), Phi_2(x)). Once the overlapping region is identified, we can construct
a inscribed circle of this region to estimate its "thickness".
Best regards，
Xu Guo