Structural topology optimization for maximum linear buckling loads by using a moving iso-surface threshold method

Abstract

This paper investigates topology design optimization for maximizing critical buckling loads of thin-walled structures using a moving iso-surface threshold (MIST) method. Formulation for maximizing linear buckling loads with additional constraints on load-path continuity and lower bound of eigenvalue is firstly presented. New physical response functions are proposed and expressed in terms of the strain energy densities determined in the two-steps of finite element buckling analysis. A novel approach by introducing a connectivity coefficient is developed to ensure continuity of effective load-path in optimum topology. The lower bound of eigenvalue is defined to eliminate spurious localized buckling modes. The MIST algorithm and its interfaces with commercial finite element (FE) software are given in detail. Numerical results are presented for topology optimization of plate-like structures to maximize critical buckling forces or displacements considering in-plane and out-of-plane buckling respectively. The FE analyses of the re-meshed final solid topologies with and without void material reveal that the presence of the void material has a significant effect on the out-of-plane buckling loads and a minor influence on the in-plane buckling loads.

Appendices

Appendix A. Calculation of strain energy density and construction of the Φ function

In MIST, the Φ function is constructed by its nodal values, which can be extracted from elemental data. The strain energy density in static stress analysis and that in buckling analysis for mode 1 in the e ^th element can be calculated by:

$$ {E}_{sd}=\frac{1}{2}{\left\{{\boldsymbol{\upsigma}}_s^e\right\}}^T\left\{{\boldsymbol{\upvarepsilon}}_s^{\mathbf{e}}\right\}=\frac{1}{2}{\left\{{\mathbf{u}}_{\mathbf{s}}^{\mathbf{e}}\right\}}^T{\left[\mathbf{B}\right]}^T\left[\mathbf{D}\right]\left[\mathbf{B}\right]\left\{{\mathbf{u}}_{\mathbf{s}}^{\mathbf{e}}\right\} $$

(A1)

$$ {E}_{\lambda d}=\frac{1}{2}{\left\{{\boldsymbol{\upsigma}}_{\lambda_1}^{\mathbf{e}}\right\}}^T\left\{{\boldsymbol{\upvarepsilon}}_{\lambda_1}^{\mathbf{e}}\right\}=\frac{1}{2{k}_{\sigma 1}}{\left\{{\mathbf{Y}}_1^{\mathbf{e}}\right\}}^T{\left[\mathbf{B}\right]}^T\left[\mathbf{D}\right]\left[\mathbf{B}\right]\left\{{\mathbf{Y}}_1^{\mathbf{e}}\right\} $$

(A2)

where {σ ^e _s } and {ε ^e _s } are the stress and strain in the static analysis; $\left\{{\boldsymbol{\upsigma}}_{\lambda_{\mathbf{1}}}^{\mathbf{e}}\right\}$ and $\left\{{\boldsymbol{\upvarepsilon}}_{\lambda_{\mathbf{1}}}^{\mathbf{e}}\right\}$ are those in buckling analysis for the 1st order mode; [B] and [D] are the strain–displacement and elastic constant matrices. The strain energy densities at Gaussian points can be determined for element by element by using (A1) and (A2), and then the Φ function surface can be constructed by evaluating their nodal values through averaging the values at the surrounding Gaussian points as in (Tong and Lin 2011; Vasista and Tong 2012) or using other stress or strain recovery techniques.

As an alternative approach, the element-based average strain energy densities for the e ^th element at static and buckling analyses can be calculated by:

$$ \begin{array}{cc}\hfill {E}_{sd}^e=\frac{1}{2{V}_e}{\left\{{u}_s^{\mathbf{e}}\right\}}^T\left[{\mathbf{k}}_{\mathbf{e}}\right]\left\{{u}_s^{\mathbf{e}}\right\}\kern2em \hfill & \hfill \left(e = 1,\ 2, \dots, {N}_e\right)\hfill \end{array} $$

(A3)

$$ \begin{array}{cc}\hfill {E}_{\lambda}^e=\frac{1}{2{k}_{\sigma 1}{V}_e}{\left\{{\mathbf{Y}}_i^e\right\}}^T\left[{\mathbf{k}}_{\mathbf{e}}\right]\left\{{\mathbf{Y}}_{\mathbf{i}}^{\mathbf{e}}\right\}\kern1em \hfill & \hfill \left(e = 1,\ 2, \dots, {N}_e\right)\hfill \end{array} $$

(A4)

where [k _e ] is the stiffness matrix of the e ^th element. The nodal values of strain energy densities can be found by using 2nd order polynomial interpolation over adjacent three elements. This alternative approach may be handy as some commercial FEA software can output data of strain energy densities at element centres or even at element nodes. Therefore, the Φ function surface is formed by connecting its nodal values and it will be further normalized to a range [−1, 1] to avoid dealing with too small or large numerals in the present computations.

Appendix B. Interfaces of MIST with MSC NASTRAN

2.1 B.1 Create FEA input file

Data input file ‘eigen.bdf for NASTRAN is created in the 1st iteration and then modified by using updated element weight factors in the subsequent iterations. In the created/modified input file, the following statements should be included in Sections ‘Global Case Control’ and ‘Bulk Data’:

Global Case Control:

Subcase 1

TITLE = static analysis

ESE(THRESH = 1.E-32) = ALL

Subcase 2

TITLE = buckling analysis

ESE(THRESH = 1.E-96) = ALL

Bulk Data:

EIGRL,1, 1.E-3,,10, 0,,,MAX

ESE statement used to directly extract elemental strain energies and their densities at each element centre. Very small threshold values are set so that the energy densities at all elements can be output. In the bulk data entry EIGRL, 1.0E-3 or the reasonable positive value must be entered in the 3rd column for satisfying the constraint of 0 < λ _min < λ ₁ to eliminate the spurious local buckling mode.

2.2 B.2 Read strain energy densities from the FEA output file

The following function can be used to read the strain energy density at element center in static analysis the 1st order mode from NASTRAN output file ‘eigen.f06’.

fid = fopen(‘eigen.f06’,‘r’);

Block = 1;

while (~feof(fid))

InputTextS = textscan(fid,‘%s’,1,‘delimiter’,‘\n’);

SS = cell2mat(InputTextS{1,1});

SS1 = ‘SUBCASE 1 * TOTAL ENERGY OF ALL ELEMENTS IN SET’;

SS2 = ‘E I G E N V A L U E A N A L Y S I S S U M M A R Y (READ ODULE)’;

TFS1 = strncmp(SS,SS1,30);

TFS2 = strncmp(SS,SS2,30);

if TFS1

InputTextS = textscan(fid,‘%s’,2,‘delimiter’,‘\n’);

HeaderLines{Block,1} = InputTextS{1};

NumCols = 4;

FormatString = repmat(‘%f’,1,NumCols);

InputTextS = textscan(fid,FormatString,46);

Data{Block,:} = cell2mat(InputTextS);

[NumRows,NumCols] = size(Data{Block});

Block = Block + 1;

elseif TFS2

break;

end

end

ESES0 = cell2mat(Data);

ESES0(:,1:3) = [];

ESES(1:Ne, :) = ESES0(1:Ne);

fclose(fid);

The following function can be used to read the strain energy density at element center in buckling analysis for the 1st order mode from NASTRAN output file ‘eigen.f06’.

fid = fopen(‘eigen.f06’,‘r’);

Block = 1;

while (~feof(fid))

InputText = textscan(fid,‘%s’,1,‘delimiter’,‘\n’);

S = cell2mat(InputText{1,1});

S1 = ‘MODE 1 * TOTAL ENERGY OF ALL ELEMENTS IN SET’;

S2 = ‘MODE 2 * TOTAL ENERGY OF ALL ELEMENTS IN SET’;

TF1 = strncmp(S,S1,30);

TF2 = strncmp(S,S2,30);

if TF1

InputText = textscan(fid,‘%s’,2,‘delimiter’,‘\n’);

HeaderLines{Block,1} = InputText{1};

NumCols = 4;

FormatString = repmat(‘%f’,1,NumCols);

InputText = textscan(fid,FormatString,45);

Data{Block,:} = cell2mat(InputText);

[NumRows,NumCols] = size(Data{Block});

Block = Block + 1;

elseif TF2

break;

end

end

ESEG = cell2mat(Data);

ESEG(:,1:3) = [];

fclose(fid);