The use of blood pumps may be necessary for people with heart problems, but there are potential risks of complications associated with this type of device, in particular hemolysis (destruction of red blood cells). Many engineering works are interested in the parametric optimization of these pumps to minimize hemolysis. In order to generalize this approach in the present work, we study the shape continuity of a coupled system of PDE modeling blood flows and hemolysis evolution in moving domains, governed respectively by non-Newtonian Navier-Stokes and by transport equations.

First, the shape continuity of the blood fluid velocity $u$ is shown. This development, which extends the one led in [1], is based on the recent progress made in [2]. Indeed, the non- Newtonian stress for blood flows can be described by the following rheological law: $$ S(Du) := (1 + |Du|)^{q−2} Du , $$ where $S(Du)$ is the stress tensor, the symmetric gradient is given by $Du := \frac{1}{2} (\nabla u + \nabla u^{\top})$, and where $q < 2$. Such fluids are called shear thinning fluids. Yet in [2], an existence result is provided for the case $q > 6/5$ in moving domains, by means of the study of Generalized Bochner spaces and the Lipschitz truncation method. Thus, these techniques are extended to the present framework of a sequence of converging moving domains.

After calculating the blood flow solutions, the velocity and stress field of the fluid are used as the coefficients for the transport equation governing the evolution of the hemolysis rate $h$: $$ \partial_t h + u \cdot \nabla h = | S(Du) |^{\gamma} (1 − h), $$ where the right hand side plays the role of a source term with saturation, for some $\gamma > 1$. From this, the shape continuity of the hemolysis rate is also proved.

Finally, these results allow to show the existence of minimum for a class of shape optimization problems based on the minimization of the hemolysis rate, in the framework of moving domains. The lack of uniqueness for shear thinning fluids solutions prevents the study of shape sensitivity from being pursued, so that an extension of this work for the purpose of computing a shape gradient must somehow consider a regularization of the present model.

[1] J. Sokolowski, J. Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains, Evol. Equ. Control Theory, 3(2):331–348, 2014.

[2] P. Nagele, M. Ruzicka. Generalized Newtonian fluids in moving domains, J. Differential Equations, 264(2):835–866, 2018.

We consider the problem of identifying an unknown portion $\Gamma$ of the boundary with a Robin condition of a d-dimensional $(d = 2;3)$ body $\Omega$ by a pair of Cauchy data $(f; g)$ on the accessible part $\Sigma$ of the boundary of a harmonic function $u$. For a fixed constant impedance $\alpha$, it is known [1] that a single measurement of $(f; g)$ on $\Sigma$ can give rise to infinitely many different domains . Nevertheless, a well-known approach to numerically solve the problem and obtain fair detection of the unknown boundary is to apply shape optimization methods. First, the inverse problem is recast into three different shape optimization formulations and the shape derivative of the cost function associated with each formulations are obtained [3]. Second, in this investigation, a new application of the so-called coupled complex boundary method – first put forward by Cheng et al. [2] to deal with inverse source problems – is presented to resolve the problem. The over-specified problem is transformed to a complex boundary value problem with a complex Robin boundary condition coupling the Cauchy pair on the accessible exterior boundary. Then, the cost function constructed by the imaginary part of the solution in the whole domain is minimized in order to identify the interior unknown boundary. The shape derivative of the complex state as well as the shape gradient of the cost functional with respect to the domain are computed. In addition, the shape Hessian at the critical point is characterized to study the ill-posedness of the problem. Specifically, the Riesz operator corresponding to the quadratic shape Hessian is shown to be compact. Also, with the shape gradient information, we devise an iterative algorithm based on a Sobolev gradient to solve the minimization problem. The numerical realization of the scheme is carried out via finite element method and is tested to various concrete example of the problem, both in two and three spatial dimensions.

[1] F. Cakoni, R. Kress. Integral equations for inverse problems in corrosion detection from partial cauchy data, Inverse Prob. Imaging, 1:229–245, 2007.

[2] X. L. Cheng, R. F. Gong, W. Han, X. Zheng. A novel coupled complex boundary method for solving inverse source problems, Inverse Problems, 30, 055002, 2014.

[3] L. Afraites, J. F. T. Rabago. Shape optimization methods for detecting an unknown boundary with the Robin condition by a single measurement, Discrete Contin. Dyn. Syst. - S, 2022. [10.3934/dcdss.2022196]

Velocity models presenting sharp interfaces are frequently encountered in seismic imaging, for instance for imaging the subsurface of the Earth in the presence of salt bodies. In order to mitigate the oversmoothing of classical regularization strategies such as the Tikhonov regularization, we propose a shape optimization approach for sharp-interface reconstructions in time-domain full waveform inversion. Using regularity results for the wave equation with discontinuous coefficients, we show the shape differentiability of the cost functional measuring the misfit between observed and predicted data, for shapes with low regularity. We propose a numerical approach based on the obtained distributed shape derivative and present numerical tests supporting our methodology.