A nonstandard free boundary problem arising in the shape optimization of thin torsion rods 

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Displacement fields

As usual, by rigid motion we mean the space R(Q) := r ∈ H1(Q;R3) : e(r) = 0 namely vector fields of the form r(x) = a + b ∧ x, with a, b ∈ R3. We define the space of Bernoulli-Navier fields BN(Q) := u ∈ H1(Q;R3) : ei j(u) = 0 ∀(i, j) = (3, 3) and the space TW(Q) := v = (vα , v3) ∈ H1(Q;R2)×L2(I;Hm1(D)) : eα β (v) = 0 ∀α , β ∈ {1, 2} . It is easy to check that, up to subtracting a rigid motion, any u ∈ BN(Q) admits the following representation:
u(x) = (ζ1(x3), ζ2(x3), ζ3(x3) − xα ζα (x3)) for some (ζα , ζ3) ∈ (Hm2(I))2 × Hm1(I)  2 L2(Q) Similarly, up to subtracting a Bernoulli-Navier field, any v ∈ TW(Q) can be written as a twist field, namely a displacement of the form v(x) = (−x2c(x3), x1c(x3), w(x)) for some c ∈ Hm1(I) , w ∈ L2(I;Hm1(D)) .
We remark that, up to rigid motions, any v ∈ TW(Q) can be written as v = u + v, with u as in (2.8) and v as in (2.9), and the decomposition is unique. We notice that the third component w of a field belonging to TW(Q) is not neces-sarily in H1(Q); nevertheless, using the representation (2.9), we see that (e13(v), e23(v)) = 1 c (x3)(−x2, x1) + ∇x w ∈ L2(Q;R2) .
Finally, exploiting Korn inequality and Poincaré-Wirtinger inequality, it is easy to show that the quotients BN(Q)/R(Q) and TW(Q)/BN(Q), endowed with the norms e33(·) L2(Ω) and ( e1,3(·) + e2,3(·) 2L2(Q))1/2 respectively, are Banach spaces.

Admissible loads

In the literature it is customary to distinguish between the stretching, bending and the torsion loads. Let us recall the definitions and fix some notations. Let F ∈ H−1(Q;R 3) be an external load. In the asymptotic procedure, it turns out that the load F enters in the limit problem with its resultant and momentum averaged on the sections. In particular, the normal component of the load to the section gives the average axial load [[F3]] , while the component lying on the section gives the average shear force [[F1]]e1 + [[F2]]e2 (for the definitions of the averages [[·]] we refer to §Nota-tions). Similarly, the normal and planar components of the average of the momentum [[x ∧ F]] give the torsion mF := [[x1F2 − x2F1]] ∈ H−1(I;R) (2.11) and the average bending moment m(b) := ([[x F x F ]];[[ x F + x F ]]) H−1(I;R2) , (2.12) F 2 3 − 3 2 −13 31 ∈ respectively. We now fix the type of exterior loads F we consider. With any Σ ∈ L2(Q;R3sym×3), that we extend to zero over R3 \ Q, we associate the distribution divΣ. As an element of H−1(Q;R3), it is characterized by divΣ, u R3 = − Q Σ · ∇u = − Q Σ · e(u) ∀ u ∈ H1(Q;R3) . (2.13) Definition 2.1.1. We say that F ∈ H−1(Q;R3) is an admissible load if it satisfies the following conditions.

asymptotics of fictitious problems

Theorem 2.3.1. As δ → 0+, the sequence C δ defined in (2.45) Γ-converges, with re-spect to the weak * topology of L∞ (Q;[0, 1]), to the limit compliance C lim defined in (2.43). In particular, for every fixed k ∈ R, the sequence φ δ (k) defined in (2.44) tends to the limit problem φ (k) given by (2.42), as δ → 0+. Proof. By definition of Γ-convergence, the statement means that the so-called Γ-liminf and Γ-limsup inequalities hold: inf liminfC δ (θ δ ) inf limsupC δ (θ δ ) : θ δ ∗ θ ≥ C lim(θ ) (2.58) : θ δ ∗ θ ≤ C lim(θ ) . (2.59)

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The proof of the main result

Proof of (2.58). Consider an arbitrary sequence θ δ ∗ θ . Let (vk, uk) ∈ TW(Q)×BN(Q) be a sequence such that C lim(θ ) = lim G, v H uk e13 vk e23 vk e33 uk )) θ dx k R3 + , R3 − ( ( ), ( ), ( . k Q j If we find a sequence ukδ ∈ H1(Q;R3) such that, for every fixed k and as δ → 0+, 1 G, ukδ R3 −→ G, vk R3 , H, ukδ R3 −→ H, uk R3 , (2.60) δ j(eδ L1 (2.61) (ukδ )) −→ j(e13(vk), e23(vk), e33(uk)) .

Table of contents :

Introduction
Notation
1 Preliminaries 
1.1 Convex Analysis and Duality Methods
1.1.1 Fenchel conjugate
1.1.2 Subgradients
1.1.3 Optimization problems
1.2 Functionals over Lp spaces
1.2.1 Integral functionals with integrand f (x, z)
1.2.2 Integral functionals with integrand g(x,u, z)
1.2.3 Fenchel conjugates for integral functionals
1.3 Γ-convergence
1.3.1 Definition and first properties
1.3.2 Relaxation
1.3.3 Convergence of minima and minimizers
1.3.4 Compactness andmetrizability
1.3.5 Mosco-convergence
1.4 Some topics inGeometricMeasureTheory
1.4.1 Functions of Bounded Variations
1.4.2 Traces
1.4.3 Cheegerproblem
1.5 Korninequalities
1.6 LinearElasticity
1.6.1 Linear elasticity
1.6.2 TheCompliance
1.6.3 Thinelastic structures: rods
1.6.4 Strainpotentials
2 Optimal design in thin rods: the small cross section limit 
2.1 Preliminaries
2.1.1 Displacement fields
2.1.2 Admissibleloads
2.1.3 Examplesof admissibleloads
2.2 Themainresult
2.3 Theproofof themainresult
2.3.1 Part I: compactness
2.3.2 Part II: asymptoticsoffictitiousproblems
2.3.3 Part III: back to the initial problems
2.4 Equivalent formulations of Φ(k) and optimality conditions
2.4.1 Equivalent formulations
2.4.2 Linkwiththe classical torsionproblem
2.4.3 Optimalityconditions
2.4.4 Looking for classical solutions for Φ(k)
2.5 Appendix
3 Optimal design in thin rods: the small volume fraction limit 
3.1 Themainresults
3.2 Theproofsof themainresults
3.3 Concentration phenomena
3.3.1 The case r = si =0
3.3.2 The case r = t =0
3.3.3 The case si =0
3.3.4 The case r =0
4 A nonstandard free boundary problem arising in the shape optimization of thin torsion rods 
4.1 Existence, uniqueness, optimality conditions, and dependence on the parameter s .
4.1.1 Primal problem
4.1.2 Dual problem
4.1.3 Properties of the map s → m(s)
4.2 LinkwiththeCheegerproblem
4.3 Existence and uniqueness of special solutions on a ball or a ring
4.4 Existence of special solutions for some other domain D
4.5 Some qualitative properties of solutions and special solutions
4.5.1 Regularity of the free boundary
4.6 Perspectives and conjectures
4.6.1 Somenumerics
4.6.2 Shapederivatives
4.7 Appendix
5 Shape derivatives for minima of integral functionals 
5.1 Preliminaries
5.1.1 Notation
5.1.2 Standingassumptions
5.1.3 Basic lemmataon integral functionals
5.2 Mainresults
5.3 Proofs
5.3.1 Appendix
5.4 Perspectives
5.4.1 Second order shapederivative
Bibliography 

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