Abstract
This article presents a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only. By employing these formulas in conjunction with a boundary integral approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This leads to significant cost savings and we demonstrate the performance on several numerical examples.
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Funding
RL and SV acknowledge support from NSF under grants DMS-1719834 and DMS-1454010. The work of SV was also supported by the Flatiron Institute (USA), a division of Simons Foundation, and by the Fondation Mathématique Jacques Hadamard (France).
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Communicated by: Silas Alben
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Appendix: Proofs
Appendix: Proofs
1.1 A.1 Proof of Lemma 1
We use the Frenet formulas (1) and associated conventions. To evaluate \({\overset {\star }{u}}{}^{\text {D}}\), we let Γ depend on the fictitious time η, setting
(where Γ stands for Γ+ or Γ−, and likewise for ℓ) and seek the relevant derivatives w.r.t. η evaluated at η = 0. Note that for η ≠ 0, s is no longer the arclength coordinate along Γη, and ∂sxη is no longer of unit norm; moreover, the length of Γη depends on η. The wall velocity U = (cℓ/L)τ for varying η is then given by
having set gη = |∂sxη| (note that g0 = 1). Our task is to evaluate d/dηUη(s) at η = 0. We begin by observing that the derivative of g is (since ∂sxη = τ and g = 1 for η = 0)
and the length ℓη of Γη and its derivative \({\overset {\star }{u}}{}^{\text {D}}\) are given (noting that s spans the fixed interval [0, ℓ] for all curves Γη) by
The last equality in (ii), which results from the assumed periodicity of θ, proves item (b) of the lemma.
Using these identities in (55), the sought derivative \({\overset {\star }{u}}{}^{\text {D}}\) of the wall velocity is found as
thus establishing item (a) of the lemma. The proof of Lemma 1 is complete.
1.2 A.2 Proof of Lemma 2
The proof proceeds by verification, and rests on evaluating divA, with the vector function A defined by
First, it is easy to check, e.g., using component notation relative to a Cartesian frame, that
Next, invoking the constitutive relation (2.2b) for both states, one has
i.e., using incompressibility
Finally, using (59) in (58a) and the balance equation in (58b), together with the corresponding identities obtained by switching (u, p) and \((\hat {\boldsymbol {u}}{},\hat {p})\), one obtains
(with the second equality stemming from \(\boldsymbol {\sigma }[\boldsymbol {u},p]\! :\!\boldsymbol {D}[\hat {\boldsymbol {u}}{}]=2\mu \boldsymbol {D}[\boldsymbol {u}]\! :\!\boldsymbol {D}[\hat {\boldsymbol {u}}{}]=2\mu \boldsymbol {D}[\hat {\boldsymbol {u}}{}]\! :\!\boldsymbol {D}[\boldsymbol {u}]=\boldsymbol {\sigma }[\hat {\boldsymbol {u}}{},\hat {p}]\! :\!\boldsymbol {D}[\boldsymbol {u}]\)). Using definitions (23a,b) of a1 and b1, we therefore observe that
The last step consists of applying the first Green identity (divergence theorem) to the above integral. The Lemma follows with the contribution of the end section ΓL therein stemming from condition (ii) in (12) and the periodicity conditions at the end sections. The latter hold by assumption for both u and \(\hat {\boldsymbol {u}}{}\), and the interior regularity of solutions in the whole channel then implies the same periodicity for ∇u and \(\boldsymbol {\nabla }\hat {\boldsymbol {u}}{}\); moreover, periodicity is also assumed for p (but not necessarily for \(\hat {p}\)) as well as for θ.
1.3 A.3 Proof of Lemmas 4 and 5
Let points x in a tubular neighborhood V of Γ be given in terms of curvilinear coordinates (s, z), so that
and let v(x) = vs(s, z)τ(s) + vn(s, z)n(s) denote a generic vector field in V. Then, at any point x = x(s) of Γ, we have
Assuming incompressibility, the condition divv = 0 can be used for eliminating ∂nvn and we obtain
Recalling now that the forward and adjoint solutions respectively satisfy u = (cℓ/L)τ and \(\hat {\boldsymbol {u}}{}=\mathbf {0}\) on Γ, and that 2D[v] = ∇v + ∇vT, we have
The corresponding stress vectors f = −pn + 2μD[u] ⋅ n and \(\hat {\boldsymbol {f}}{}=-\hat {p}\boldsymbol {n}+2\mu \boldsymbol {D}[\hat {\boldsymbol {u}}{}]\! \cdot \!\boldsymbol {n}\) on Γ are found as
which in particular prove items (c) of Lemmas 4 and 5. Finally, using the above in (64) establishes the remaining items (a), (b) of both lemmas.
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Bonnet, M., Liu, R. & Veerapaneni, S. Shape optimization of Stokesian peristaltic pumps using boundary integral methods. Adv Comput Math 46, 18 (2020). https://doi.org/10.1007/s10444-020-09761-7
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DOI: https://doi.org/10.1007/s10444-020-09761-7