2022/03/04 by Karol Bołbotowski, Tomasz Lewiński
Engineering · #Composite Material Mechanics #Elasticity and Material Modeling #Topology Optimization in Engineering
paper · pdf · doi:10.1007/s00526-022-02186-8
openalex publication_date 2022/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Abstract The paper deals with the Free Material Design (\text FMD) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mtext>FMD</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:math> problem aimed at constructing the least compliant structures from an elastic material, the constitutive field of which plays the role of design variable in the form of a tensor valued measure λ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math> supported in the design domain. Point-wise the constitutive tensor is referred to a given anisotropy class \mathscr H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math> , while the integral of a cost c(λ ) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo>(</mml:mo><mml:mi>λ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is bounded from above. The convex p -homogeneous elastic potential j is parameterized by the constitutive tensor. The work puts forward the existence result and shows that the original problem can be reduced to the Linear Constrained Problem (\mathrm LCP) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>LCP</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> known from the theory of optimal mass distribution by G. Bouchitté and G. Buttazzo. A theorem linking solutions of (\text FMD) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mtext>FMD</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:math> and (\mathrm LCP) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>LCP</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> allows to effectively solve the original problem. The developed theory encompasses several optimal anisotropy design problems known in the literature as well as it unlocks new ones. By employing the derived optimality conditions we give several analytical examples of optimal designs.