European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS35: Large-scale Biomechanics: Models, Solvers and HPC (Part 2)
Chair: Lorenzo Zanon
10:40
25 mins
Reduced order models for the simulation of microfluidic devices for biological fluids
George Biros, Gokberk Kabacaoglu
Abstract: We will discuss fast solvers, reduced order models, and optimization methods for fluid-structure interaction problems in the zero Reynolds number regime. Our goal is to design a deterministic lateral displacement (DLD) device to sort same-size biological cells by their deformability, in particular to sort red blood cells (RBCs) by their viscosity contrast between the fluid in the interior and the exterior of the cells. A DLD device optimized for efficient cell sorting enables rapid medical diagnoses of several diseases such as malaria since infected cells are stiffer than their healthy counterparts. In this context we will describe an integral equation formulation that delivers optimal complexity solvers for this type of problems. Despite its excellent theoretical properties, our integral equation solver remains prohibitively expensive for optimization and uncertainty quantification. We will summarize our efforts to reduce the computational costs, starting from low-resolution discretization, domain decomposition, and high-dimensional approximation. High-dimensional approximation, which we do using multilayer perceptrons, is used to construct surrogate models not for the whole solver---but for the action of specific nonlinear operators. The final scheme blends ultra low-resolution solvers (who on their own cannot resolve the flow), several regression multilayer perceptrons, and an operator time-stepping scheme, which we introduced to specifically enable the use of surrogate models. We have used our methodology successfully for flows that are completely different from the flows in the training dataset.
11:05
25 mins
Computational fluid dynamics in cerebral aneurysms: our experience
Helena Švihlová, Jaroslav Hron
Abstract: The cerebral aneurysm is a local extension of a brain vessel with high prevalence in population (about 20%). The aneurysm rupture is the worst form of a stroke with high rate of mortality and severe morbidity (about 66%). On the other hand, treatment of unruptured aneurysms is an active and controversial topic since the aneurysm rupture is rare and the preventive surgery can also cause the damages. Next to the growing amount of neuroimaging examinations leading to an aneurysm detection, there is no precise method to predict the rate of aneurysm growth as well as the risk of rupture. [1] It is well known that hemodynamics has a significant role in an aneurysm development. Computational fluid dynamics (CFD) in cerebral aneurysms is currently a hot research topic because it can provide some of the hemodynamic parameters in a patient-specific manner. In fact, even the ‘AHA/ASA Guidelines for Management of Aneurysmal Subarachnoid Hemorrhage’ now recommends clinicians to ‘consider hemodynamic characteristics of the aneurysm when discussing the risk of aneurysm rupture’. [2] The mostly discussed hemodynamic parameters are wall shear stress (WSS), the magnitude of the stress vector acting tangencially on the vessel wall, and oscillatory shear index, the scalar value representing wall shear stress vector oscillations during a cardiac cycle. In 2017, a meta-analysis was published which, based on 1257 aneurysms from 22 studies, evaluated differences in hemodynamic parameters between ruptured and unruptured aneurysms. It showed that in ruptured aneurysms, WSS is significantly lower than in unruptured.[3] On the other hand, the largest ever study comparing ruptured and unruptured aneurysms has shown that wall shear stress is higher in ruptured aneurysms, than in unruptured aneurysms.[4] Another important factor affecting IA hemodynamics is their size. In our own study, we evaluated morphological and hemodynamic parameters in 20 mid-cerebral aneurysms. We compared aneurysms according to 2 parameters: rupture status (ruptured versus unruptured) and size (small <10mm and large >10mm). It was shown that many hemodynamic parameters in small aneurysms were statistically significantly different, independent of its rupture status. In our contribution, we will present assumptions and limitations of the models CFD uses as well as the potential benefits in a clinical practice in a near future. We will discuss the limitations at different levels of the blood flow modelling and our experience with setup of computational methods using academic open source computational tools (PETSc,FEniCS, VMTK). Attention may also be paid to image segmentation and boundary condition extraction as well as non-newtonian behaviour of the blood and vessel wall movement in a brain. We will briefly discuss the numerical methods in context of high performance computing and we will put CFD research in context of modern imaging methods development. [1] Seibert B, Tummala RP, Chow R, Faridar A, Mousavi SA, Divani AA. Intracranial aneurysms: review of current treatment options and outcomes. Front Neurol. 2011; 2:45 [2] Connolly E et al. Guidelines for the management of aneurysmal subarachnoid hemorrhage a guideline for healthcare professionals from the american heart association/american stroke association. Stroke. 2012; 43(6): 1711–1737. [3] Zhou G, Zhu Y, Yin Y, Su M, Li M. Association of wall shear stress with intracranial aneurysm rupture: systematic review and meta-analysis. Sci Rep 2017;7(1):5331 [4] Cebral JR, Mut F, Weir J, Putman C. Quantitative characterization of the hemodynamic environment in ruptured and unruptured brain aneurysms. AJNR Am J Neuroradiol 2011;32(1):145-151
11:30
25 mins
Reduced Order Methods for PDEs: state of the art and perspectives with applications in cardiovascular flows
Gianluigi Rozza
Abstract: We provide the state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs), and we focus on some perspectives in their current trends and developments, with a special interest in parametric problems arising in offline-online Computational Fluid Dynamics (CFD) and applications for cardiovascular flows Efficient parametrizations (random inputs, geometry, physics) are very important to be able to properly address an offline-online decoupling of the computational procedures and to allow competitive computational performances. Current ROM developments in CFD include: a better use of stable high fidelity methods to enhance the quality of the reduced model too; more efficient sampling techniques to reduce the number of the basis functions, retained as snapshots, as well as the dimension of online systems [3]; the improvements of the certification of accuracy based on residual based error bounds and of the stability factors, as well as the the guarantee of the stability of the approximation with proper space enrichments, and reduction in parameter space. For nonlinear systems, also the investigation on bifurcations of parametric solutions are crucial and they may be obtained thanks to a reduced eigenvalue analysis of the linearised operator. All the previous aspects are very important in CFD problems to focus in real time on complex parametric biomedical flow problems, even in a flow control setting. Model flow problems will focus on few benchmarks, as well as on simple fluid-structure interaction problems [1, 2], instabilities and bifurcations due to Coanda effect in Mitral valves regurgitation [4, 5], as well as flows in carotid arteries properly parametrised to allow patient specific simulations [6]. References [1] F. Ballarin and G. Rozza, “POD–Galerkin monolithic reduced order models for parametrized fluid-structure interaction problems”, International Journal for Numerical Methods in Fluids, 82(12), p. pp. 1010–1034, 2016. [2] F. Ballarin, G. Rozza, and Y. Maday, “Reduced-order semi-implicit schemes for fluid-structure interaction problems”, in Model Reduction of Parametrized Systems, P. Benner, M. Ohlberger, A. Patera, G. Rozza, and K. Urban (eds.), Springer International Publishing, vol. 17, p. pp. 149–167, 2017. [3] M. Hess, A. Alla, A. Quaini, G. Rozza, and M. Gunzburger, “A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions”, 2018. [4] M. Hess, A. Quaini, and G. Rozza, “Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature”, 2019. [5] G. Pitton, A. Quaini, and G. Rozza, “Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology”, Journal of Computational Physics, 344, p. pp. 534–557, 2017. [6] M. Tezzele, F. Ballarin, and G. Rozza, “Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods”, in Mathematical and Numerical Modeling of the Cardiovascular System and Applications, D. Boffi, L. F. Pavarino, G. Rozza, S. Scacchi, and C. Vergara (eds.), Cham: Springer International Publishing, p. pp. 185–207, 2018.