Parallel solvers for virtual element discretizations of elliptic equations in mixed form

The aim of this paper is twofold. On the one hand, we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

BPX preconditioners for the Bidomain model of electrocardiology

The aim of this work is to develop a BPX preconditioner for the Bidomain model of electrocardiology. This model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction\u2013diffusion partial differential equation (PDE) and an elliptic linear PDE, modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations, the so-called membrane model, describing the ionic currents through the cellular membrane. The discretization of the coupled system by finite elements in space and semi-implicit finite differences in time yields at each time step the solution of an ill-conditioned linear system. The goal of the present study is to construct, analyze and numerically test a BPX preconditioner for the linear system arising from the discretization of the Bidomain model. Optimal convergence rate estimates are established and verified by two- and three-dimensional numerical tests on both structured and unstructured meshes. Moreover, in a full heartbeat simulation on a three-dimensional wedge of ventricular tissue, the BPX preconditioner is about 35% faster in terms of CPU times than ILU(0) and an Algebraic Multigrid preconditioner

A numerical investigation on the use of the virtual element method for topology optimization on polygonal meshes

A classical formulation of topology optimization addresses the problem of finding the best distribution of an assigned amount of isotropic material that minimizes the work of the external forces at equilibrium. In general, the discretization of the volume-constrained minimum compliance problem resorts to the adoption of four node displacement-based finite elements, coupled with element-wise density unknowns. When regular meshes made of square elements are used, well-known numerical instabilities arise, see in particular the so-called checkerboarded patterns. On the other hand, when unstructured meshes are needed to cope with geometry of any shape, additional instabilities can steer the optimizer towards local minima instead of the expected global one. Unstructured meshes approximate the strain energy of the members of the arising optimal design with an accuracy that is strictly related to the geometrical features of the discretization, thus remarkably affecting the achieved layouts. In light of the above remarks, in this contribution we consider polygonal meshes and implement the virtual element method (VEM) to solve two classes of topology optimization problems. The robustness of the adopted discretization is exploited to address problems governed by (nearly incompressible and compressible) linear elasticity and problems governed by Stokes equations. Numerical results show the capabilities of the proposed polygonal VEM-based approach with respect to more conventional discretizations

VEM and topology optimization on polygonal meshes

Topology optimization is a fertile area of research that is mainly concerned with the automatic generation of optimal layouts to solve design problems in Engineering. The classical formulation addresses the problem of finding the best distribution of an isotropic material that minimizes the work of the external loads at equilibrium, while respecting a constraint on the assigned amount of volume. This is the so-called minimum compliance formulation that can be conveniently employed to achieve stiff truss-like layout within a two-dimensional domain. A classical implementation resorts to the adoption of four node displacement-based finite elements that are coupled with an elementwise discretization of the (unknown) density field. When regular meshes made of square elements are used, well-known numerical instabilities arise, see in particular the so-called checkerboard patterns. On the other hand, when unstructured meshes are needed to cope with geometry of any shape, additional instabilities can steer the optimizer towards local minima instead of the expected global one. Unstructured meshes approximate the strain energy of truss-like members with an accuracy that is strictly related to the geometrical features of the discretization, thus remarkably affecting the achieved layouts. In this paper we will consider several benchmarks of truss design and explore the performance of the recently proposed technique known as the Virtual Element Method (VEM) in driving the topology optimization procedure. In particular, we will show how the capability of VEM of efficiently approximating elasticity equations on very general polygonal meshes can contribute to overcome the aforementioned mesh-dependent instabilities exhibited by classical finite element based discretization technique

A Parallel Preconditioner for 2D Elliptic Boundary Value Problems

This work presents the implementation on a Linux Cluster of a parallel preconditioner for the solution of the linear system resulting from the finite element discretization of a 2D second order elliptic boundary value problem. The numerical method, proposed by Bramble, Pasciak and Schatz, is developed using Domain Decomposition techniques, which are based on the splitting of the computational domain into subregions of smaller size, enforcing suitable compatibility conditions. The Fortran code is implemented using PETSc: a suite of data structures and routines devoted to the scientific parallel computing and based on the MPI standard for all message-passing communications. The main interest of the paper is to investigate how the architectural aspects of the cluster influence the performance of the considered algorithm. We provide an analysis of the execution times as well as of the scalability, using as test case the classical Poisson equation with Dirichlet boundary conditions

Role of infarct scar dimensions, border zone repolarization properties and anisotropy in the origin and maintenance of cardiac reentry

Cardiac ventricular tachycardia (VT) is a life-threatening arrhythmia consisting of a well organized structure of reentrant electrical excitation pathways. Understanding the generation and maintenance of the reentrant mechanisms, which lead to the onset of VT induced by premature beats in presence of infarct scar, is one of the most important issues in current electrocardiology. We investigate, by means of numerical simulations, the role of infarct scar dimension, repolarization properties and anisotropic fiber structure of scar tissue border zone (BZ) in the genesis of VT. The simulations are based on the Bidomain model, a reaction-diffusion system of Partial Differential Equations, discretized by finite elements in space and implicit-explicit finite differences in time. The computational domain adopted is an idealized left ventricle affected by an infarct scar extending transmurally. We consider two different scenarios: i) the scar region extends along the entire transmural wall thickness, from endocardium to epicardium, with the exception of a BZ region shaped as a central sub-epicardial channel (CBZ); ii) the scar region extends transmurally along the ventricular wall, from endocardium to a sub-epicardial surface, and is surrounded by a BZ region (EBZ). In CBZ simulations, the results have shown that: i) the scar extent is a crucial element for the genesis of reentry; ii) the repolarization properties of the CBZ, in particular the reduction of IKs and IKr currents, play an important role in the genesis of reentrant VT. In EBZ simulations, since the possible reentrant pathway is not assigned a-priori, we investigate in depth where the entry and exit sites of the cycle of reentry are located and how the functional channel of reentry develops. The results have shown that: i) the interplay between the epicardial anisotropic fiber structure and the EBZ shape strongly affects the propensity that an endocardial premature stimulus generates a cycle of reentry; ii) reentrant pathways always develop along the epicardial fiber direction; iii) very thin EBZs rather than thick EBZs facilitate the onset of cycles of reentry; iv) the sustainability of cycles of reentry depends on the endocardial stimulation site and on the interplay between the epicardial breakthrough site, local fiber direction and BZ rim

AC joint osteoarthritis: the role of genetics. An MRI evaluation of asymptomatic elderly twins

Purpose: The anatomy of the articular surfaces has historically identified as major responsible for acromioclavicular joint osteoarthritis (ACJO). On the other side, the almost 100% prevalence of ACJO in subjects over 50 years old seems to suggest a multifactorial etiology. We compared ACJO between asymptomatic elderly monozygotic (MZ) and dizygotic (DZ) twins to investigate the influence of genetics and environmental factors. Materials and Methods: Thirty pairs of twins [15MZ-15DZ; mean age (SD): 63.70 (3.31); range: 53–72] were retrospectively enrolled. ACJO was evaluated on MRI through a 4-grade severity scale and ACJ configuration was assessed. Information regarding work activity were obtained. Heritability index was calculated. Results: The intraclass correlation coefficient (ICC) value of 0.868 (95% CI; 0.798 to 0.917). An ICC values of 0.889 (95% CI; 0.798 to 0.944) and 0.843 (95% CI, 0.712 to 0.920) were found in the MZ and DZ groups, respectively. The polychoric correlation was 0.857 in the MZ twins and 0.757 in the DZ twins. The calculated heritability index was 0.20 (20%), and the contribution of the shared environment (c2) and unique environment (e2) was 0.66 (66%) and 0.14 (14%), respectively. No relationship between job types and ACJO in both the total cohort (r = 0.089; p = 0.499) and in the monozygotic (r = 0.247; p = 0.187) and the dizygotic twin groups (r = −0.084; p = 0.658) was found. Conclusions: The role of genetics on ACJO accounts for only 20%; a specific anatomical configuration of the articular surfaces only partially acts on the development of joint osteoarthritis. Environmental factors have the greatest impact. Level of Evidence: IV

Obesity and bone loss at menopause: The role of sclerostin

Background. Peripheral fat tissue is known to positively influence bone health. However, evidence exists that the risk of non-vertebral fractures can be increased in postmenopausal women with obesity as compared to healthy controls. The role of sclerostin, the SOST gene protein product, and body composition in this condition is unknown. Methods. We studied 28 severely obese premenopausal (age, 44.7 \ub1 3.9 years; BMI, 46.0 \ub1 4.2 kg/m2 ) and 28 BMI-matched post-menopausal women (age, 55.5 \ub1 3.8 years; BMI, 46.1 \ub1 4.8 kg/m2 ) thorough analysis of bone density (BMD) and body composition by dual X-ray absorptiometry (DXA), bone turnover markers, sclerostin serum concentration, glucose metabolism, and a panel of hormones relating to bone health. Results. Postmenopausal women harbored increased levels of the bone turnover markers CTX and NTX, while sclerostin levels were non-significantly higher as compared to premenopausal women. There were no differences in somatotroph, thyroid and adrenal hormone across menopause. Values of lumbar spine BMD were comparable between groups. By contrast, menopause was associated with lower BMD values at the hip (p < 0.001), femoral neck (p < 0.0001), and total skeleton (p < 0.005). In multivariate regression analysis, sclerostin was the strongest predictor of lumbar spine BMD (p < 0.01), while menopausal status significantly predicted BMD at total hip (p < 0.01), femoral neck (p < 0.001) and total body (p < 0.05). Finally, lean body mass emerged as the strongest predictor of total body BMD (p < 0.01). Conclusions. Our findings suggest a protective effect of obesity on lumbar spine and total body BMD at menopause possibly through mechanisms relating to lean body mass. Given the mild difference in sclerostin levels between pre-and postmenopausal women, its potential actions in obesity require further investigation

A numerical study of scalable cardiac electro-mechanical solvers on HPC architectures

We introduce and study some scalable domain decomposition preconditioners for cardiac electro-mechanical 3D simulations on parallel HPC (High Performance Computing) architectures. The electro-mechanical model of the cardiac tissue is composed of four coupled sub-models: (1) the static finite elasticity equations for the transversely isotropic deformation of the cardiac tissue; (2) the active tension model describing the dynamics of the intracellular calcium, cross-bridge binding and myofilament tension; (3) the anisotropic Bidomain model describing the evolution of the intra- and extra-cellular potentials in the deforming cardiac tissue; and (4) the ionic membrane model describing the dynamics of ionic currents, gating variables, ionic concentrations and stretch-activated channels. This strongly coupled electro-mechanical model is discretized in time with a splitting semi-implicit technique and in space with isoparametric finite elements. The resulting scalable parallel solver is based on Multilevel Additive Schwarz preconditioners for the solution of the Bidomain system and on BDDC preconditioned Newton-Krylov solvers for the non-linear finite elasticity system. The results of several 3D parallel simulations show the scalability of both linear and non-linear solvers and their application to the study of both physiological excitation-contraction cardiac dynamics and re-entrant waves in the presence of different mechano-electrical feedbacks