Wolff's law based topology optimization for continuum structure

Wolff's law in biomechanics states that the bone continuallyadapts to its mechanical environment through cell-based remodeling oftrabecular surfaces and the local microstructure tends to align with theprincipal directions of the stress. The objective of the present research is todevelop a new rule-based method for continuum topology optimization based onWolff's law. The major ideas of the present approach are as follows.Firstly, the structure is to be optimized as a piece of bone whichobeys Wolff's law. Secondly, the process of finding the optimal structuraltopology is equivalent to the ``bone'' remodeling/growth process. Thirdly,the remodeling rule can be explained as follows: During the process ofgrowth, at any material point in the structure, if the absolutevalues of one of its principal strains is out of a given interval of referencestrain, then the material in the local microstructure along the correspondingdirection should be adjusted. If theabsolute values of all its principal strains locate in the interval, the material point is in a state of equilibrium of remodeling.Finally, the global optimization of structure requires all material pointsto be in the state of remodeling equilibrium under the loading conditions.In order to express the microstructure and the anisotropic behavior of amaterial point, a second rank positive and definite fabric tensor isintroduced. The relative density of a point in design domain expressed bythe invariants of the fabric tensor through the mathematical condensation ofthe porous medium based on the stiffness-equivalence rule is used to displaythe optimal topology of structure. Examples are given to show the validityand capability of the proposed approach for the optimal topology design ofcontinuum structures.