Interpolation/penalization applied for strength design of 3D thermoelastic structures

Abstract

With design independent loads and only a constrained volume (no local bounds), the same optimal design leads simultaneously to minimum compliance and maximum strength. However, for thermoelastic structures this is not the case and a maximum volume may not be an active constraint for minimum compliance. This is proved for thermoelastic structures by sensitivity analysis of compliance that facilitates localized determination of sensitivities, and the compliance is not identical to the total elastic energy (twice strain energy). An explicit formula for the difference is derived and numerically illustrated with examples. In compliance minimization for thermoelastic structures it may be advantageous to decrease the total volume, but for strength maximization it is argued to keep the total permissible volume. Linear interpolation (no penalization) may to a certain extent be argued for 2D thickness optimized designs, but for 3D design problems interpolation must be included and not only from the penalization point of view to obtain 0–1 designs. Three interpolation types are presented in a uniform manner, including the well known one parameter penalizations, named SIMP and RAMP. An alternative two parameter interpolation in explicit form is preferred, and the influence of interpolation on compliance sensitivity analysis is included. For direct strength maximization the sensitivity analysis of local von Mises stresses is demanding. An applied recursive procedure to obtain uniform energy density is presented in details, and it is shown by examples that the obtained designs are close to fulfilling also strength maximization. Explicit formulas for equivalent thermoelastic loads in 2D and 3D finite element analysis are derived and applied, including the sensitivity analysis.

Appendices

Appendix A: Equivalent loads from temperature

According to textbooks the load case $\{A_{\epsilon_0}\}$ equivalent to initial strains {ε ₀} is

$$ \{A_{\epsilon_0\}} = \int_V [B]^T [L] \{\epsilon_0\} d V $$

(27)

where [B]^T is the transposed strain/displacement matrix, [L] is the constitutive matrix and integration is over a FE element volume V, i.e., the analysis is restricted to a specific element. To deal with matrices and vectors of smaller order (27) is written on the directional level (i = x, y, z). Furthermore, the present paper is restricted to elements where $[B]^T [L] \{\epsilon_0\}$ is constant in the element (omitting element index e)

$$ \{A_{\epsilon_0}\}_i = \int_V [B]_i^T [L] \{\epsilon_0\} d V = [B]_i^T [L] \{\epsilon_0\} V $$

(28)

For an isotropic 3D model we have in traditional notation

$$ \begin{array}{rll} [L] &=& \frac {E (1 - \nu)}{(1 + \nu)(1 - 2 \nu)}\\&&\!\times \left[ \begin{array}{c c c c c c} 1 & \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 0 & 0 & 0 \\ \frac {\nu}{1 - \nu} & 1 & \frac {\nu}{1 - \nu} & 0 & 0 & 0 \\ \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1 - 2 \nu}{2(1 - \nu)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1 - 2 \nu}{2(1 - \nu)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1 - 2 \nu}{2(1 - \nu)} \end{array} \right] \end{array} $$

(29)

where E is Young’s modulus of elasticity and ν is Poisson’s ratio.

For initial strain due to a uniform temperature change ΔT in the element

$$ \{\epsilon_0\} = \alpha \Delta T \{1, 1, 1, 0, 0, 0\}^T $$

(30)

where α is the thermal expansion coefficient. Then the product [L] {ϵ ₀} in (28) is

$$ [L] \{\epsilon_0\} = \frac {E \alpha \Delta T}{1 - 2 \nu} \{1, 1, 1, 0, 0, 0\}^T $$

(31)

Appendix B: Element geometry parameters and load distributions

For a tetrahedron element (omitting element index e) an arbitrary three dimensional Cartesian coordinate system has x, y, z axes. The element geometries are described by the corner point positions that also serve as nodal points in the FE model. Relative positions are used to reduce the number of parameters, and areas and volumes are expressed directly in these relative positions.

Figure 7 shows the most simple tetrahedron finite element with linear displacement assumption, parameterized by the parameters p ₁ − p ₉. Element translation has no influence. The nominal element volume V is determined by

$$ \begin{array}{rll} V &=& \frac {1}{6} (p_1p_5p_9 - p_1p_6p_8 + p_2p_6p_7 \\ &&\quad- p_2p_4p_9 + p_3p_4p_8 - p_3p_5p_7) > 0 \end{array} $$

(32)

with enforced positive condition by the order of the four nodes.

Fig. 7

Definition of parameters p ₁ − p ₉, that describes the tetrahedron geometry. The order of the four nodes I,II,III,IV is restricted by p ₁ p ₅ p ₉ − p ₁ p ₆ p ₈ + p ₂ p ₆ p ₇ − p ₂ p ₄ p ₉ + p ₃ p ₄ p ₈ − p ₃ p ₅ p ₇ > 0

For this simple tetrahedron element the displacement assumption for a displacement v _i in the i-direction is

$$ v_i = \{H\}^T [K]^{-1} \{D\}_i ~~~ \text {with} ~~~ \{H\}^T = \left\{ 1 ~~~ \frac {x}{h} ~~~ \frac {y}{h} ~~~ \frac {z}{h}\right\} $$

(33)

where the vector {D} _i contains the nodal displacements in direction i, h is a reference length, and for the [K]^− 1 matrix see Pedersen (2006). From this follows that the strain/displacement sub-matrix in the x-direction are

$$ \begin{array}{rll} [B]_x^T &=& [K]^{-T} \left [ \frac {\partial \{H\}}{\partial x} ~~ \{0\} ~~ \{0\} ~~ \frac {\partial \{H\}}{\partial y} ~~ \frac {\partial \{H\}}{\partial z} ~~ \{0\} \right ] \\ & =& \frac {1}{6 V} \left [ \begin{array}{c c c c} 6 V & -q_x h & -q_y h & -q_z h \\ 0 & p_{5968}h & p_{6749}h & p_{4857}h \\ 0 & p_{3829}h & p_{1937}h & p_{2718}h \\ 0 & p_{2635}h & p_{3416}h & p_{1524}h \end{array} \right ]\\ &&\times\left [ \begin{array}{c c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1/h & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/h & 0 & 0 \\ 0 & 0 & 0 & 0 & 1/h & 0 \end{array} \right ] \\ & =&\frac {1}{6 V} \left [ \begin{array}{c c c c c c} -q_x & 0 & 0 & -q_y & -q_z & 0 \\ p_{5968} & 0 & 0 & p_{6749} & p_{4857} & 0 \\ p_{3829} & 0 & 0 & p_{1937} & p_{2718} & 0 \\ p_{2635} & 0 & 0 & p_{3416} & p_{1524} & 0 \end{array} \right ] \end{array} $$

(34)

where V is the nominal volume (a solid tetrahedron). With definition of a short notation, exemplified by

$$ p_{5968} := p_5p_9 - p_6p_8 $$

(35)

and additional three practical parameters are defined

$$ \begin{array}{rll} q_x &=& p_{5968} + p_{3829} + p_{2635} \\ q_y &=& p_{6749} + p_{1937} + p_{3416} \\ q_z& =& p_{4857} + p_{2718} + p_{1524} \end{array} $$

(36)

Note, that the sums of each column two, three and four of matrix [K]^− T are all zero, which finally give force equilibrium.

Similarly for the $[B]_y^T$ and the $[B]_z^T$ sub-matrices we get

$$ \begin{array}{rll} [B]_y^T & =& [K]^{-T} \left [\{0\} ~~ \frac {\partial \{H\}}{\partial y} ~~ \{0\} ~~ \frac {\partial \{H\}}{\partial x} ~~ \{0\} ~~ \frac {\partial \{H\}}{\partial z} \right ] \\ & =& \frac {1}{6 V} \left [ \begin{array}{c c c c c c} 0 & -q_y & 0 & -q_x & -q_z & 0 \\ 0 & p_{6749} & 0 & p_{5968} & p_{4857} & 0 \\ 0 & p_{1937} & 0 & p_{3829} & p_{2718} & 0 \\ 0 & p_{3416} & 0 & p_{2635} & p_{1524} & 0 \end{array} \right ] \end{array} $$

(37)

$$ \begin{array}{rll} [B]_z^T & =& [K]^{-T} \left [\{0\} ~~ \{0\} ~~ \frac {\partial \{H\}}{\partial z} ~~ \{0\} ~~ \frac {\partial \{H\}}{\partial x} ~~ \frac {\partial \{H\}}{\partial y} \right ] \\ & =& \frac {1}{6 V} \left [ \begin{array}{c c c c c c} 0 & 0 & -q_z & 0 & -q_x & -q_y \\ 0 & 0 & p_{4857} & 0 & p_{5968} & p_{6749} \\ 0 & 0 & p_{2718} & 0 & p_{3829} & p_{1937} \\ 0 & 0 & p_{1524} & 0 & p_{2635} & p_{3416} \end{array} \right ] \end{array} $$

(38)

For a 3D isotropic tetrahedron model (34), (31) inserted in (28) result in the temperature load nodal distribution for the x-direction expressed by the simple formula

$$ \{A_{\Delta T}\}_x = \frac {f_t(\rho) E \alpha \Delta T}{1 - 2 \nu} \left \{ \begin{array}{c} -q_x \\ p_{5968} \\ p_{3829} \\ p_{2635} \end{array} \right \} $$

(39)

where the load is interpolated with the factor f _t (ρ) as applied in (20). When (37) is applied in stead of (34) we similarly get for the y-direction

$$ \{A_{\Delta T}\}_y = \frac {f_t(\rho) E \alpha \Delta T}{1 - 2 \nu} \left \{ \begin{array}{c} -q_y \\ p_{6749} \\ p_{1937} \\ p_{3416} \end{array} \right \} $$

(40)

and when (38) is applied for the z-direction

$$ \{A_{\Delta T}\}_z = \frac {f_t(\rho) E \alpha \Delta T}{1 - 2 \nu} \left \{ \begin{array}{c} -q_z \\ p_{4857} \\ p_{2718} \\ p_{1524} \end{array} \right \} $$

(41)

which are large forces that are very sensitive to the value of Poisson’s ratio ν.

Similar formula for 2D simple triangles are listed in Pedersen and Pedersen (2010), but there assuming linear interpolation.

Appendix C: Elastic energy density, compliance density and difference

The intention of Appendix C is to clarify the difference between elastic energy and compliance for thermoelastic problems, it is not essential for the performed optimizations. The involved quantities refer to a point in space, i.e., a material point.

The elastic energy density u (twice strain energy density with linear elasticity) is for a thermoelastic problem defined by

$$ u = (\{\widetilde \epsilon\}^T - \{\epsilon_0\}^T) [L] (\{\widetilde \epsilon\} - \{\epsilon_0\}) $$

(42)

where $\{\widetilde \epsilon\}$ contains the “non physical” strains directly corresponding to a resulting displacement field as described by {D}, {ϵ ₀} contains the initial strains from a temperature increase ΔT as described in Appendix A, and [L] is the constitutive matrix for an isotropic case. With actual components as in Appendix A

$$ \begin{array}{rll} && \{\widetilde \epsilon\} = \left \{ \begin{array}{c} \widetilde \epsilon_{11} \\ \widetilde \epsilon_{22} \\ \widetilde \epsilon_{33} \\ 2 \widetilde \epsilon_{12} \\ 2 \widetilde \epsilon_{13} \\ 2 \widetilde \epsilon_{23} \end{array} \right \}, ~~~ \{\epsilon_0\} = \alpha \Delta T \left \{ \begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right \}, \\ && [L] \{\epsilon_0\} = \frac {E \alpha \Delta T}{1 - 2 \nu} \left \{ \begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right \} \end{array} $$

(43)

Expanding the expression (42) give

$$ u = \{\widetilde \epsilon\}^T [L] \{\widetilde \epsilon\} + \{\epsilon_0\}^T [L] \{\epsilon_0\} - 2 \{\widetilde \epsilon\}^T [L] \{\epsilon_0\} $$

(44)

The first term is named compliance density ϕ

$$ \phi = \{\widetilde \epsilon\}^T [L] \{\widetilde \epsilon\} $$

(45)

using the name compliance density because a compliance Φ can be obtained by integration over the continuum/structural volume V as $\{\widetilde \epsilon\}$ is evaluated directly from the displacement field {D}.

$$ \Phi = \int_V \phi d V = \int_V \{\widetilde \epsilon\}^T [L] \{\widetilde \epsilon\} d V = \{D\}^T [S] \{D\} $$

(46)

The difference between the elastic energy density u and the compliance density ϕ can be evaluated as a scalar since

$$ \begin{array}{l} \{\epsilon_0\}^T [L] \{\epsilon_0\} = 3 \frac {E \alpha^2 \Delta T^2}{1 - 2 \nu}, \\ \{\widetilde \epsilon\}^T [L] \{\epsilon_0\} = (\widetilde \epsilon_{11} + \widetilde \epsilon_{22} + \widetilde \epsilon_{33}) \frac {E \alpha \Delta T}{1 - 2 \nu} \end{array} $$

(47)

which give

$$ u - \phi = \frac {E \alpha \Delta T}{1 - 2 \nu} (3 \alpha \Delta T - 2 (\widetilde \epsilon_{11} + \widetilde \epsilon_{22} + \widetilde \epsilon_{33})) $$

(48)

and alternatively the sum of the normal strains could be expressed in a sum of normal stresses.