Michell truss type theories as a Γ-limit of optimal design in linear elasticity

A Derivation of the variational form of the compliance minimization problem

Here we repeat basically our presentation from [23, Section 2.4]. We include this part in order to keep the present article self-contained.

Let g ∈ W - 1 , 2 ⁢ ( Ω ¯ ; ℝ n ) . The aim of this appendix is to give a derivation of the compliance minimization problem in its variational form

where the infimum is taken over the set

Here the equation - div ⁡ σ = g is to be understood as an equation in the distributional sense in a neighborhood of Ω ¯ , with σ extended by 0 on the complement of Ω. In this way we incorporate boundary conditions in the equation; see our discussion in Section 1.1. We want to derive this variational problem starting from the standard formulation of a linear elasticity problem. More details can be found in [2].

Consider Ω ⊂ ℝ n as an elastic body, characterized by its elasticity tensor A 0 ∈ Lin ⁢ ( ℝ sym n × n ; ℝ sym n × n ) , where for simplicity we assume here that A 0 = Id ℝ sym n × n is the identity. We remove a subset H ⊂ Ω from the elastic body and the new boundaries from that process shall be traction-free. The resulting linear elasticity problem is to find u : Ω ∖ H → ℝ n such that

where e ⁢ ( u ) = 1 2 ⁢ ( ∇ ⁡ u + ∇ ⁡ u ) T . The compliance (work done by the load) is given by

where u : Ω ∖ H → ℝ 2 is the unique solution to the linear elasticity system above. We want to minimize the compliance under a constraint on the “weight” ℒ 2 ⁢ ( Ω ∖ H ) . We do so by the introduction of a Lagrange multiplier λ, and are interested in the minimization problem

The “equivalence” between the mass constrained problem and the problem including a Lagrange multiplier only holds on a heuristic level; see [17, 18, 19] for a discussion of this point. Accepting this step, taking the limit of vanishing weight corresponds to the limit λ → ∞ . We now rewrite the problem by considering space-dependent elasticity tensors of the form A ⁢ ( x ) = χ ⁢ ( x ) ⁢ A 0 , where χ ∈ L ∞ ⁢ ( Ω ; { 0 , 1 } ) . The equations from above turn into the system

The compliance turns into a functional on the set of permissible elasticity tensors, and is given by

where u is the solution of (A.2). By the principle of minimum complementary energy, the compliance can be written as

where

and σ ∈ L ∞ ⁢ ( Ω ; ℝ sym n × n ) is a solution of the PDE

i.e., σ ∈ S g ⁢ ( Ω ) . We see that the compliance minimization problem can be understood as the variational problem of finding the infimum

Of course, the compliance of a pair ( χ , σ ) is infinite if there exists a set of positive measure U such that χ = 0 and σ ≠ 0 on U. Hence the above variational problem is equivalent with

where

Up to a factor λ - 1 / 2 , this is just the integrand (3.5), and hence (A.3) is just the variational problem (A.1), with A 0 = Id ℝ sym n × n . As is well known, this problem does not possess a solution in general and requires relaxation.

B A very brief presentation of Michell trusses

In this section, we want to sketch very briefly how the limit integral functional 𝒢 ∞ , g is linked to Michell truss theory for the case n = 2 . What we say here is mainly taken from [9].

A truss is a finite union of bars (line segments that can resist compression or tension parallel to them) between points x i ∈ ℝ 2 , i = 1 , … , M . We write ( x 1 , … , x m ) = x ∈ ℝ 2 × M , and let w ∈ ℝ sym M × M . To every bar [ x i , x j ] = { t ⁢ x i + ( 1 - t ) ⁢ x j : t ∈ [ 0 , 1 ] } , we associate w i ⁢ j , where | w i ⁢ j | is the strength of the bar, and the sign of w i ⁢ j is chosen according to whether the bar has to withstand compression or tension.

The force, provided the bar [ x i , x j ] , is given by

The set of all bars shall counterbalance a given force

where y i , g i ∈ ℝ 2 , i = 1 , … , N , are given. The truss ( { x i } i = 1 , … , M , { w i ⁢ j } i , j = 1 , … , M ) withstands g if

The weight of the truss ( x , w ) is given by

The task is now the minimization of the weight, given the external forces, as a function of x , w . To express how this variational problem relates to 𝒢 ∞ , g , we note that the force supplied by the bars can be written as the divergence of a stress, f i ⁢ j ⁢ ( x , w ) = div ⁡ σ i ⁢ j ⁢ ( x , w ) with

With this notation, the balance of forces becomes the equation

and the weight of the truss is given by 𝒲 ⁢ ( x , w ) = | σ ⁢ ( x , w ) | (the total variation of the measure σ).

Summarizing, we are dealing with the variational problem

To guarantee the existence of a minimizer, this variational problem requires relaxation, as has already been remarked by Michell in 1904 [22]. We will not discuss the derivation of the relaxation here and refer the interested reader to [9]. We only state the result: Namely, that it becomes the variational problem defined by the Γ-limit in the main text:

Requiring additionally supp ⁡ σ ⊂ Ω ¯ leads to