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Simultaneous discrete and continuum multiresolution topology optimization

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Abstract

In the field of continuum structures, the density-based methods for topology optimization are well known and broadly studied. Likewise, in the area of (quasi-) optimal discrete element structures, there is significant work which can even consider nonlinear constitutive models. This work seeks to set a precedent by combining these strategies, in other words, to topology optimize continuum and discrete elements simultaneously with the possibility of including nonlinear constitutive models for the discrete elements. Reinforced concrete, reinforced masonry, fiber-reinforced materials, rib-reinforced shell structures, and others are problems which are conveniently modeled using a combination of both, discrete and continuum elements. Thus, a combined and simultaneous framework to topologically optimize these type of hybrid structures breaks down the barrier often separating both fields. The simultaneous optimization of continuum and discrete poses several mathematical and numerical challenges, some of which have been previously documented. The present work addresses a large number of these challenges and presents a robust and stable computational implementation as a proof-of-concept.

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Notes

  1. This problem analogous to stress recovery problem in the finite element analysis of solids, where various techniques make an attempt to better approximate the knowingly smoother field.

  2. Concave regions and holes, when present, need to be identified and dealt with using restriction zones (Zegard and Paulino 2014)

  3. This may result in a volume constraint violation, but the optimization process will rapidly address this situation.

  4. The reinforcing steel does contribute in the compressive behavior of reinforced concrete. However, in the scope of topology optimization of discrete and continuum elements, we guide the optimization to use of discrete steel elements towards their key role: support the tension zones in reinforced concrete.

Abbreviations

\(\textbf{a}\) :

Truss cross-sectional areas vector

\(A_{\text {se}}\) :

Area associated with the s-th density of the e-th parent element

\(a_i\) :

Truss cross-sectional areas of the i-th element

\(a_i^{(\min )},\,a_i^{(\max )}\) :

Lower and upper bounds, respectively, on the i-th cross-sectional area

\(\textbf{B}\) :

Strain–displacement matrix

c :

Objective function

\(\textbf{D}\) :

Element’s constitutive matrix

\(\det {\textbf{J}_{\text {se}}}\) :

Determinant (or scale factor) relating the infinitesimal areas in the cartesian and the subelement natural coordinate systems

\(E_\text {OG}\) :

Ogden tangent modulus

\(E_\text {SV}\) :

Saint-Venant tangent modulus

\(E_\text {NH}\) :

Neo-Hookean tangent modulus

\(E_\text {HK}\) :

Henky tangent modulus

\(E_\text {TO}\) :

Tension-only tangent modulus

\(E_i\) :

Elastic modulus of the i-th continuum element

\(E_{\min }\) :

Elastic modulus of the void phase

\(E_0\) :

Elastic modulus of the solid phase

\(E_c^{\prime }\) :

Concrete’s Young modulus

\(E_s\) :

Steel’s Young modulus

\(\textbf{f}\) :

Nodal force vector

f :

Volume fraction of the solid phase

\(\textbf{H}\) :

Filter matrix

J :

Jacobian

\(\textbf{K}\) :

Global stiffness matrix

\(\textbf{K}_0^{\left( i\right) }\) :

Stiffness matrix of the i-th element calculated with a unit elastic modulus (i.e., \(E=1\))

\(\textbf{K}_{\text {cont}}\) :

Continuum’s global stiffness matrix

\(\textbf{K}_{\text {truss}}\) :

Global linearized discrete (truss) stiffness matrix

\(\textbf{K}^{(i)}\) :

Tangent stiffness matrix in global coordinates for the i-th member

\(\textbf{K}^{(i)}_{\text {truss-local}}\) :

Tangent stiffness matrix in local coordinates for the i-th member

\(\textbf{l}\) :

Truss element lengths vector

\(\ell _i\) :

Truss length of the i-th element

\(\ell _0\) :

Truss element initial length

\(N_{\text {cont}}\) :

Number of continuum elements

\(N_s\) :

Number of density subelements

\(N_{\text {truss}}\) :

Number of truss elements

p :

Penalization exponent associated with the SIMP power law

\(\textbf{r}\) :

Residual force vector

\(r_{\min }\) :

Filter radius

\(\textbf{t}\) :

Internal force vector

\(t_{\text {se}}\) :

Thickness associated with the s-th density of the e-th parent element

\(\textbf{u}\) :

Nodal displacement vector

U :

Internal structure energy

\(\textbf{v}\) :

Elements’ volumes vector

\(V_0\) :

Design domain volume

\(V_\text {truss}^\text {(lim)}\) :

Maximum available structural volume

\(w_i,\,w_j\) :

Weights associated with the Gauss quadrature rule

\(\textbf{x}\) :

Element’s design variable vector

\(\textbf{x}^\text {cont}\) :

Continuum elements design variables vector

\(\textbf{x}^\text {truss}\) :

Relaxed binary (continuous) design varibable associated with the truss elements vector

\(x_i^\text {cont}\) :

Design variable of the i-th continuum element

\(x_i^\text {truss}\) :

Design variable of the i-th truss element

\(\beta\) :

Ogden model parameter

\(\gamma\) :

Ogden model parameter

\(\lambda\) :

Stretch

\(\varvec{\xi },\,\varvec{\xi }_s\) :

Natural or intrinsic local coordinates associated with the element and the subelement, respectively

\(\varPi\) :

Total potential energy of the system

\(\varPi _{\min }\) :

Stationary potential energy of the system

\(\varvec{\rho }\) :

Elements’ densities vector

\(\rho _i\) :

Density of the i-th (finite) element

\(\rho _{\text {{se}}}\) :

Density associated with the s-th density of the e-th parent element

\(\sigma _\text {OG}\) :

Ogden Cauchy’s stress

\(\sigma _\text {SV}\) :

Saint-Venant Cauchy’s stress

\(\sigma _\text {NH}\) :

Neo-Hookean Cauchy’s stress

\(\sigma _\text {HK}\) :

Henky Cauchy’s stress

\(\sigma _\text {TO}\) :

Tension-only Cauchy’s stress

\(\varPsi _j\) :

Specific strain energy of the i-th element

\(\varPsi _\text {OG}\) :

Ogden specific strain energy

\(\varPsi _\text {SV}\) :

Saint-Venant specific strain energy

\(\varPsi _\text {NH}\) :

Neo-Hookean specific strain energy

\(\varPsi _\text {HK}\) :

Henky specific strain energy

\(\varPsi _\text {TO}\) :

Tension-only specific strain energy

\(\varOmega _e\) :

The domain of the e-th element

\(\varOmega\) :

Total potential of the loads

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Acknowledgements

The authors want to thank Diego Salinas for his revisions, feedback, suggestions and comments throughout the development of this work.

Funding

This work received financial support from the Chilean government agency responsible for coordinating, promoting, and aiding scientific research as part of the project FONDEF ID17I20264.

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Authors and Affiliations

  1. Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, 7820436, Santiago, Chile

    Gonzalo Mejías & Tomás Zegard

Authors
  1. Gonzalo Mejías
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  2. Tomás Zegard
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Correspondence to Tomás Zegard.

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Replication of results

The numerical results presented in this document can be replicated using the methodology and formulations described here. The base meshes used in the examples are available upon request to the authors.

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Responsible Editor: Xiaojia Shelly Zhang

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Mejías, G., Zegard, T. Simultaneous discrete and continuum multiresolution topology optimization. Struct Multidisc Optim 66, 137 (2023). https://doi.org/10.1007/s00158-023-03592-y

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