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A dimension reduction procedure for the design of lattice-spring systems with minimal fabrication cost and required multi-functional properties

Egor Makarenkov, Sakshi Malhotra, Yang Jiao

TL;DR

This work tackles the problem of designing elastoplastic lattice-spring networks with multi-functional requirements while minimizing fabrication cost. By applying dimension reduction via inequalities, the authors reduce the design space for a small 4-spring lattice to a analytically tractable form, enabling explicit evaluation of cost and performance. They identify Case 9 as the optimal topology, computing an exact solution with $C_{min} \approx 2.077$ at $(c_1,c_2,c_3,c_4) \approx (0.75,0.577,0.577,0.173)$ and showing all other configurations cannot surpass this value. The results provide a practical analytic pathway for topology optimization in elastoplastic lattice systems with current-conducting springs, yielding explicit fabrication parameters and rigorous justification via reduced-variable analyses and appendix derivations.

Abstract

We show that the problem of the design of the lattices of elastoplastic current conducting springs with optimal multi-functional properties leads to an analytically tractable problem. Specifically, focusing on a lattice with a small number of springs, we use the technique of inequalities to reduce the number variables and to compute the minimal cost of lattice fabrication explicitly.

A dimension reduction procedure for the design of lattice-spring systems with minimal fabrication cost and required multi-functional properties

TL;DR

This work tackles the problem of designing elastoplastic lattice-spring networks with multi-functional requirements while minimizing fabrication cost. By applying dimension reduction via inequalities, the authors reduce the design space for a small 4-spring lattice to a analytically tractable form, enabling explicit evaluation of cost and performance. They identify Case 9 as the optimal topology, computing an exact solution with at and showing all other configurations cannot surpass this value. The results provide a practical analytic pathway for topology optimization in elastoplastic lattice systems with current-conducting springs, yielding explicit fabrication parameters and rigorous justification via reduced-variable analyses and appendix derivations.

Abstract

We show that the problem of the design of the lattices of elastoplastic current conducting springs with optimal multi-functional properties leads to an analytically tractable problem. Specifically, focusing on a lattice with a small number of springs, we use the technique of inequalities to reduce the number variables and to compute the minimal cost of lattice fabrication explicitly.
Paper Structure (15 sections, 2 theorems, 61 equations, 10 figures, 11 tables)

This paper contains 15 sections, 2 theorems, 61 equations, 10 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Formulation of the optimization problem
  3. Reduction to 2 variables and analytic computation of the minimal cost $C_{min}$ for Case 9.
  4. Case 9
  5. Reduction to 1 or 2 variables and non-reachability of the cost of $C_{min}$ for all other cases.
  6. Case 1.
  7. Case 2.
  8. Case 3.
  9. Case 4.
  10. Case 5.
  11. Case 6.
  12. Case 7.
  13. Case 8.
  14. Case 10.
  15. Appendix

Key Result

Proposition 1

If $(c_{1*},c_{2*})$ solves reduced problem (opt9reduced) then solves full problem (opt9).

Figures (10)

  • Figure 1: Left: Visualization of the two-variable estimate for Case 9.1, Right: Two-variable estimate for Case 9.2.
  • Figure 2: Visualization of the single variable estimate for Case 1.
  • Figure 3: Vizualization of the single variable estimate for Case 2.
  • Figure 4: Vizualization of the single variable estimate for Case 3 with $x=c_1+c_3$.
  • Figure 5: Left: Visualization of the two-variable estimates for Case 4, Right: Left Figure zoomed in.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2