As 'compliance' is something positive, shouldn't it be maximized instead of minimize?
Hello,
In mechanics, "compliance" roughly stands for the inverse of "stiffness". To improve the structural integrity you may often want to minimize the strains in the structure which means minimizing the compliance or maximizing the stiffness. For some other applications, such as in designing morphing structures, you may want to increase compliance at certain regions. Therefore it is not really correct to regard "compliance" something negative or positive as it might be desirable or undesirable depending on the aim.
Best,
Gokhan
Dear Shahroz,
Topological optimization opens up new possibilities for a more rational design. For its part, the architect and structural engineers develops and is inspired by optimal structural forms that often resemble those that can be found in nature.
Previous studies have reduced computation time from a minimum compliance formulation to a minimum volume formulation and leveraging the benefits of rigid preconditioning in a reanalysis-based procedure.
Throughout this design process, the engineer can impose limits so that the design uses the resources optimally while remaining physically viable.
For more details and information about this subject, i suggest you see links in topic.
-topology optimization algorithms for the solution of compliance ... - Cimat
www.cimat.mx/~ivvan/.../TopologyOptimization/MAOM_Thesis.p...
-Compliance-topology optimization - OptiStruct - Altair HyperWorks ...
https://forum.altairhyperworks.com › ... › OptiStruct
-compliance minimization using topology optmization method ... - UFRN
arquivos.info.ufrn.br/.../COMPLIANCE_MINIMIZATION_USIN...
-Minimum Compliance Topology Optimization... (PDF Download ...
https://www.researchgate.net/.../319059921_Minimum_Complianc...
-Topology optimization for minimum compliance under multiple loads ...
https://link.springer.com/article/10.1007/s00158-007-0214-3
-Methodology for Topology and Shape Optimization ... - Chalmers
https://www.chalmers.se/.../Methodology_for_Topology_and_Sha...
Best regards
As Gokhan has noted, minimizing compliance simply means making the optimized model as stiff as possible overall, or, in other words, keeping the total strain energy to a minimum (numerically, think of compliance as twice the total strain energy). Whether this is desirable or to what degree is a question the designer has to answer. Note also that while compliance is an overall concept, minimizing compliance does not mean you are minimizing the maximum displacement at a given point or keeping it within certain bounds. Usually there are also the stresses to worry about.
it's also my problem and this " minimizing the compliance " is confusing me and I don't understand why we use this minimize and also what does compliance mean here?
1. as others mentioned, it is a practical problem with interests.
2. i guess, because minimizing compliance is probably the easiest structural optimization problem to sovle. it is self adjoint. you do not even need to solve an extra adjoint problem to do the sensitivity analysis.
I know what you say but it is kind of confusing ......I guess we should just accept it and use in our method ...........
The word "compliance" is based on the word "comply" as in "comply with". "To comply" basically means to bend to (or to act in accordance with) a wish or order. So, in the structural engineering sense, "compliance" can be interpreted as overall flexibility under loading. As a result, minimizing compliance for a structure is the same as maximizing the overall stiffness of the structure. However, note that, as I have written earlier, this does not necessarily mean that you have minimized the maximum displacement at a critical point. Numerically, compliance is simply the equivalent load vector times the displacement vector (or an integration of the stress times strain tensors), and therefore it is a measure of the flexibility only in an average sense. After minimization of compliance, parts of the structure may still show unacceptable displacements, while other parts become very stiff. I hope this is helpful.
One more item: When optimizing the structure under multiple load cases, each load case will lead to a different compliance value, so this has to be taken into account.
In structural optimization problems, we have usually two kinds of objective function: local for example the displacement in specified point and global one as the strain energy or compliance for overall the structure. The latter is more convenient for using the calculus of variations or the control theory. Minimization of compliance leads to the minimum strain energy stored in the structure to carry out loads acting on it.
All comments give an explanation about the correspondence between the minimization of the compliance of a structure and its stiffness.
I would not give further digressions since they are quite clear and well written.
However I think that the question proposed by Shahroz Khan regarding the maximization of the compliance instead of its minimization has not been answered yet. Shahroz Khan states he finds unclear why we minimize the compliance "since it is something positive".
I would stress that there is no relation between the sign of the goal functional (here, the compliance) and the kind of the stationarity condition (maximum or minimum).
Even in standard one-variable analysis one can search for minima (or maxima) of a positive (or negative) function.
Think about the search for the shape of a component having the minimum weight (ensuring structural integrity): it is a well-posed problem though the involved objective functional (the mass) is a positive quantity.
The mentioned link between minimal compliance and maximal stiffness of a structure is not general: To obtain the stiffest design it depends on the loads and BCs, if a minimization or maximization is to be performed, which what I think is what the original question was aiming at.
Since compliance is calculated as c=f^T * u (f transpose times u) there are mainly three possibilities:
- an external force vector is applied, f=const. Then minimizing compliance means minimizing u at the points of load application -> stiffest design.
- loads are applied as prescribed displacements, u=const. Then maximizing compliance means, that the reaction forces are maximized -> stiffest design
- mixed forces and prescribed displacements: Its not trivial to tell if minimizing or maximizing needs to be performed for the stiffest design.
As Olaf Ambrozkiewicz indicates, thinking of "minimisation of compliance" as obtaining the stiffest design is a bit "dangerous" as it depends on the type of loading. See the study of Klarbring and Stromberg on this topic (Article Topology optimization of hyperelastic bodies including non-z...
). My coauthors and I utilised this property for the design of flexures, seeArticle A simple and versatile topology optimization formulation for...
Consider reading the "old" work of William Prager (1968) on optimal design; here the author defines the 'static elastic compliance', see Article Optimality criteria in structural design
for one load case and linear elastic material and only mass constraint... global compliance is a convex function (cf a paper from K. Svanberg (Article Local and Global Optima
)) that it is a useful mathematical property for optimisation algorithms convergence.
one of the objective functions in topology optimization could be the improvement of stiffness which is explained by the inverse of Compliance in mechanics.
The formula for Stiffness = (1/Compliance). Hence wen you minimize the compliance, you maximize the stiffness of the object you are working, which would be the case most of the time. Example: You perform topology optimization and remove materials, you might consider having maximum stiffness while removing materials. Software's like Hypermesh has an objective known as "minimize compliance" which is ideally to maximize the stiffness.
This is ideally the case for leaf springs in precision mechanisms too. But if you are safe under the von misses stress, you can play with compliance if movements are necessary. (Examples: You try to bend a pen, hence you apply UDL load on both edges of pen). These types in detail are known as flexure designs where you need compliance of the system also to be high. Check "design of flexures" in google. In real life they are used in watch mechanisms.
The Stiffness equal (1/Compliance), so when you minimize the compliance, you maximize the stiffness of the system you are designing.
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