Implicit Heaviside filter with high continuity based on suitably graded THB splines

Abstract

The variable density topology optimization (TO) method has been applied to various engineering fields because it can effectively and efficiently generate the conceptual design for engineering structures. However, it suffers from the problem of low continuity resulting from the discreteness of both design variables and explicit Heaviside filter. In this paper, an implicit Heaviside filter with high continuity is introduced to generate black and white designs for TO where the design space is parameterized by suitably graded truncated hierarchical B-splines (THB). In this approach, the fixed analysis mesh of isogeometric analysis is decoupled from the design mesh, whose adaptivity is implemented by truncated hierarchical B-spline subjected to an admissible requirement. Through the intrinsic local support and high continuity of THB basis, an implicit adaptively adjusted Heaviside filter is obtained to remove the checkboard patterns and generate black and white designs. Threefold advantages are attained in the proposed filter: a) The connection between analysis mesh and adaptive design mesh is easily established compared with the traditional adaptive TO method using nodal density; b) the efficiency in updating design variables is remarkably improved than the traditional implicit sensitivity filter based on B-splines under successive global refinement; and c) the generated black and white designs are preliminarily compatible with current commercial computer aided design system. Several numerical examples are used to verify the effectiveness of the proposed implicit Heaviside filter in compliance and compliant mechanism as well as heat conduction TO problems.

Graphical abstract

Keywords

topology optimization / truncated hierarchical B-spline / isogeometric analysis / black and white designs / Heaviside filter

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Nomenclature

Abbreviations
CAD	Computer aided design
FEM	Finite element method
HB	Hierarchical B-splines
IGA	Isogeometric analysis
ITO	Isogeometric topology optimization
NURBS	Nonuniform rational B-splines
SIMP	Solid isotropic material with penalization
TO	Topology optimization
THB	Truncated hierarchical B-splines
Variables
A_e^l	Volume of the eth element of level l
A_j	Volume of the jth active element
alpha	Coefficient to control the descent speed of the gradient of design variables
B	B-spline basis function space
B_{\mathrm{A}}^{l,\mathrm{n}\mathrm{e}\mathrm{w}}	New added active basis functions of level l
B_{\mathrm{A},\mathrm{r}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}	The active functions to be refined
B_{\mathrm{A},\mathrm{u}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}	Active functions remaining unaltered
c	Compliance
c_k^{l+1}	Coefficient of Bl with respect to B_k^{l+1}
c_k^{\tau ,l+1}	Truncated coefficient of Bl with respect to B_k^{l+1} for standard HB
\widetilde{c_k^{\tau ,l+1}}	Truncated coefficient of Bl with respect to B_k^{l+1} for simplified HB
{\mathit{C}}_{\mathrm{A}}^{N-1,\mathrm{n}\mathrm{e}\mathrm{w}}	Transformation matrix to map the new function space into basis functions of level N − 1
{\mathit{C}}_{\mathrm{A}}^{N-1,\mathrm{o}\mathrm{l}\mathrm{d}}	Transformation matrix to map the old function space into basis functions of level N − 1
{\mathit{C}}_l	Transformation matrix to map the active THB basis functions up to level l to the basis functions of level l
{\mathit{C}}_l^{l+1}	Transformation matrix to map {\mathit{B}}_{\mathrm{A}}^{l,\mathrm{n}\mathrm{e}\mathrm{w}} to {\mathit{B}}_{\mathrm{A},\mathrm{r}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}
C({\mathit{B}}_i^l)	Children of {\mathit{B}}_k^{l+1} in Bl+1
C(Q)	Children of Q
D_i^k	Updating factor
E	Elastic modulus
E_0	Elastic modulus of solid material
E_{min}	Elastic modulus of void material
f	External force
f_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}	Input force
J_{\mathrm{u}}^l	Index transformation matrix to obtain the indices of {\mathit{B}}_{\mathrm{A},\mathrm{u}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}} in {\mathit{B}}_{\mathrm{A}}^{l,\mathrm{n}\mathrm{e}\mathrm{w}}
kinput, kout	Input and output springs, respectively
K	Stiffness matrix
{\mathit{K}}_e	Stiffness matrix of the eth element
{\mathit{K}}_j^0	Stiffness matrix of the jth quadrature points
ke	Number of active elements affected by R_{\mathrm{A}}^i
kq	Number of quadrature points in the region supporting of R_{\mathrm{A}}^i
m	Admissible parameter
move	Move limit
M_{nd}	Gray coefficient describes the design converged to a black and white (0/1) discrete solution
{\mathcal{M}}_{\text{c}}	Union of deactivated elements to be reactivated
{\mathcal{M}}_{\text{r}}	Union of elements to be refined
N	Number of levels of the THB basis function space
N^l	Dimension of spline space of level l
N_{\text{c}}^l	Number of active control points of level l
N_e^l	Number of active elements of level l
{\mathcal{N}}_{\mathrm{c}}(\mathcal{Q},Q,m)	Coarsening neighborhood of Q under m admissible constraint
{\mathcal{N}}_{\mathrm{r}}(\mathcal{Q},Q,m)	Refinement neighborhood of Q under m admissible constraint
NF	Number of density coefficients
p	Degree of B-spline
penal	Penalty factor
P({\mathit{B}}_k^{l+1})	Parent of {\mathit{B}}_i^l in Bl
P(Q)	Parent of Q
Q	Cartesian mesh
R_0, R_1	Active basis functions of levels 0 and 1, respectively
R_{\mathrm{A}}	An arbitrary active THB basis function
R_{\mathrm{A}}^i(\xi ,\eta )	Value of the ith active basis function at parameter point (\xi ,\eta )
R_l	Active basis functions of level l
S(Q,k)	Support extension of Q with respect to level k
T	Temperature
tl	Thermal load
truncl+1	Truncated operation
\widetilde{\text{trun}{\text{c}}^{l+1}}\left({\mathit{B}}_i^l\right)	Truncated operation for simplified HB basis
u	Displacement
u_{\mathrm{o}\mathrm{u}\mathrm{t}}	Output displacement
{\mathit{u}}_e	Displacement of the eth element
u(x)	Nodal displacement vector
V	Volume of the design domain
\overline{\mathit{V}}	Upper limit of the volume fraction for the solid material
V(x)	Material volume
x	Union of material density design variables of all active control points
x_{\text{G},j}	Density value of the jth Gaussian quadrature points
α	Value of an active THB basis function
Ξ	Knot vectors
Ω	Parametric domain
\mathcal{H}	Standard HB basis
\mathcal{Q}	Hierarchical mesh
\tilde{\mathcal{H}}	Simplified HB basis
\mathcal{T}	Standard THB basis
\tilde{\mathcal{T}}	Simplified THB basis
ρ	Density
{\rho }_{\mathrm{A}}	Density design variable of the active control point of R_{\mathrm{A}}
{\rho }_{\mathrm{A}}^i	ith density design variable
{\rho }^{\mathrm{o}\mathrm{l}\mathrm{d}}, {\rho }^{\mathrm{n}\mathrm{e}\mathrm{w}}	Original and locally refined density fields, respectively
{\rho }_{\mathrm{c}}^{\mathrm{n}\mathrm{e}\mathrm{w}}	Locally coarsened density field
{\rho }_{\mathrm{A},\mathrm{r}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}	Design variables with respect to B_{\mathrm{A},\mathrm{r}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}
{\rho }_{\mathrm{A},\mathrm{u}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}	Design variables with respect to B_{\mathrm{A},\mathrm{u}}^{l,\mathrm{o}\mathrm{l}\mathrm{d}}
{\rho }_e^l	Centroid density of the eth active element of level l
{\rho }_{\mathrm{G}}^j	Density of the jth quadrature points in the locally supported region
{\rho }_i^k	ith density design variables at the kth iterative steps
{\rho }_{\mathrm{l}\mathrm{o}\mathrm{w}}, {\rho }_{\mathrm{u}\mathrm{p}}	Minimum and maximum values of the densities to determine the blurry regions of the design domain, respectively
{\rho }_{min}	Minimum value of the densities in the coarsening updating rule
\rho (\xi ,\eta )	Density at an arbitrary parameter coordinate (\xi ,\eta )
\nu	Poisson’s ratio
β	An arbitrary B-spline basis function
η	Damping factor
λ	Lagrange multiplier
μ	Shift parameter

Acknowledgements

This work was supported by the National Key R&D Program of China (Grant No. 2020YFB1708300) and China Postdoctoral Science Foundation (Grant No. 2021M701310).

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