Isogeometric topology optimization of compliant mechanisms using transformable triangular mesh (TTM) algorithm

Abstract

This paper presents a unique solution to the problem of planar compliant mechanism design by means of geometric morphing technology and isogeometric analysis (IGA). A new transformable triangular mesh (TTM) component is developed based on geometric morphing technology, which can generate the required topology with different feature sets from a surface that has zero boundary component (interior hole) under the control of Laplace energy and mesh operations. Such flexible TTM component is helpful in overcoming the initial dependency of conventional topology optimization methods in which the layout of parameterized components often affects final optimized results. As the high-order continuity between the grids of IGA can improve calculation accuracy and numerical stability, IGA is combined with the presented TTM algorithm to establish a two-layer computational model so as to identify the optimal compliant mechanism topology within a given design domain and given displacements of input and output ports. In the upper layer of the model, the compliant limbs are characterized explicitly by triangular grids. By moving, splitting, and refining these triangular grids, the generated shape will then be projected onto the lower layer which is discretized using NURBS elements so as to calculate structural sensitivity for driving new iteration. To demonstrate the benefits provided by such method for compliant mechanism design, several numerical studies are tested, in which the geometry freely evolves along the optimization procedure, resulting in more efficient non-trivial topologies with desired kinematic behavior.

Appendix

For the ith interior non-control vertex, its Laplacian energy can be expressed:

$$ {\displaystyle \begin{array}{c}{E}_i=\sum \limits_{j\in N}{w}_{ij}{||\left({\boldsymbol{v}}_i^{\prime }-{\boldsymbol{v}}_j^{\prime}\right)-{\boldsymbol{R}}_i\left({\boldsymbol{v}}_i-{\boldsymbol{v}}_j\right)||}^2\\ {}\kern1.4em =\sum \limits_{j\in N}{w}_{ij}{\left({\boldsymbol{e}}_{ij}^{\prime }-{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}\right)}^{\mathrm{T}}\left({\boldsymbol{e}}_{ij}^{\prime }-{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}\right)\\ {}\kern1.4em =\sum \limits_{j\in N}{w}_{ij}\left({{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}{\boldsymbol{e}}_{ij}^{\prime }-2{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}+{\boldsymbol{e}}_{ij}^{\mathrm{T}}{\boldsymbol{e}}_{ij}\right)\end{array}} $$

(33)

In order to minimize energy function E_i, we only need to minimize the non-constant terms with R_i in E_i. Thus, the minimization of the Eq. (33) can be written as:

$$ {\displaystyle \begin{array}{c}\underset{{\boldsymbol{R}}_i}{\mathrm{argmin}}\sum \limits_{j\in N}-{w}_{ij}2{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}=\underset{{\boldsymbol{R}}_i}{\mathrm{argmax}}\kern0.3em \sum \limits_{j\in N}{w}_{ij}{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}\\ {}\kern4.399998em =\underset{{\boldsymbol{R}}_i}{\mathrm{argmax}}\kern0.5em Tr\left(\sum \limits_{j\in N}{w}_{ij}{\boldsymbol{R}}_i{\boldsymbol{e}}_{ij}{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}\right)\\ {}\kern4.399998em =\underset{{\boldsymbol{R}}_i}{\mathrm{argmax}}\kern0.5em Tr\left({\boldsymbol{R}}_i\sum \limits_{j\in N}{w}_{ij}{\boldsymbol{e}}_{ij}{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}\right)\end{array}} $$

(34)

Furthermore, we introduce the covariance matrix:

$$ {\boldsymbol{S}}_i=\sum \limits_{j\in N}{w}_{ij}{\boldsymbol{e}}_{ij}{{\boldsymbol{e}}^{\prime}}_{ij}^{\mathrm{T}}={\boldsymbol{P}}_i{\boldsymbol{D}}_i{{\boldsymbol{P}}^{\prime}}_i^{\mathrm{T}} $$

(35)

where D_i is a diagonal matrix that contains the weights w_ij. P_i is a 2 × N(j) matrix containing e_ij’s as its columns, and it is similar for P_i^′. When R_iS_i is symmetric positive semi-definite, the rotation matrix R_i that maximizes Tr (R_iS_i) can be obtained. R_i can be derived from the singular value decomposition (SVD) of ${\boldsymbol{S}}_i={\boldsymbol{U}}_i{\Sigma}_i{\boldsymbol{V}}_i^{\mathrm{T}}$:

$$ {\boldsymbol{R}}_i={\boldsymbol{V}}_i{\boldsymbol{U}}_i^{\mathrm{T}} $$

(36)

up to changing the sign of the column of U_i that corresponds to the smallest singular value ensuring det (R_i) > 0.