58 research outputs found
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations
The focus of this study is the construction and numerical validation of
parallel block preconditioners for low order virtual element discretizations of
the three-dimensional Maxwell equations. The virtual element method (VEM) is a
recent technology for the numerical approximation of partial differential
equations (PDEs), that generalizes finite elements to polytopal computational
grids. So far, VEM has been extended to several problems described by PDEs, and
recently also to the time-dependent Maxwell equations. When the time
discretization of PDEs is performed implicitly, at each time-step a large-scale
and ill-conditioned linear system must be solved, that, in case of Maxwell
equations, is particularly challenging, because of the presence of both H(div)
and H(curl) discretization spaces. We propose here a parallel preconditioner,
that exploits the Schur complement block factorization of the linear system
matrix and consists of a Jacobi preconditioner for the H(div) block and an
auxiliary space preconditioner for the H(curl) block. Several parallel
numerical tests have been perfomed to study the robustness of the solver with
respect to mesh refinement, shape of polyhedral elements, time step size and
the VEM stabilization parameter.Comment: 21 pages, 10 tables, 4 figure
A parallel solver for FSI problems with fictitious domain approach
We present and analyze a parallel solver for the solution of fluid structure
interaction problems described by a fictitious domain approach. In particular,
the fluid is modeled by the non-stationary incompressible Navier-Stokes
equations, while the solid evolution is represented by the elasticity
equations. The parallel implementation is based on the PETSc library and the
solver has been tested in terms of robustness with respect to mesh refinement
and weak scalability by running simulations on a Linux cluster.Comment: Contribution to the 5th African Conference on Computational Mechanic
Parallel inexact Newton-Krylov and quasi-Newton solvers for nonlinear elasticity
In this work, we address the implementation and performance of inexact
Newton-Krylov and quasi-Newton algorithms, more specifically the BFGS method,
for the solution of the nonlinear elasticity equations, and compare them to a
standard Newton-Krylov method. This is done through a systematic analysis of
the performance of the solvers with respect to the problem size, the magnitude
of the data and the number of processors in both almost incompressible and
incompressible mechanics. We consider three test cases: Cook's membrane
(static, almost incompressible), a twist test (static, incompressible) and a
cardiac model (complex material, time dependent, almost incompressible). Our
results suggest that quasi-Newton methods should be preferred for compressible
mechanics, whereas inexact Newton-Krylov methods should be preferred for
incompressible problems. We show that these claims are also backed up by the
convergence analysis of the methods. In any case, all methods present adequate
performance, and provide a significant speed-up over the standard Newton-Krylov
method, with a CPU time reduction exceeding 50% in the best cases
A parallel solver for fluid structure interaction problems with Lagrange multiplier
The aim of this work is to present a parallel solver for a formulation of
fluid-structure interaction (FSI) problems which makes use of a distributed
Lagrange multiplier in the spirit of the fictitious domain method. The fluid
subproblem, consisting of the non-stationary Stokes equations, is discretized
in space by - finite elements, whereas the
structure subproblem, consisting of the linear or finite incompressible
elasticity equations, is discretized in space by finite
elements. A first order semi-implicit finite difference scheme is employed for
time discretization. The resulting linear system at each time step is solved by
a parallel GMRES solver, accelerated by block diagonal or triangular
preconditioners. The parallel implementation is based on the PETSc library.
Several numerical tests have been performed on Linux clusters to investigate
the effectiveness of the proposed FSI solver.Comment: 27 pages, 8 figures, 14 table
BDDC preconditioners for virtual element approximations of the three-dimensional Stokes equations
The Virtual Element Method (VEM) is a novel family of numerical methods for
approximating partial differential equations on very general polygonal or
polyhedral computational grids. This work aims to propose a Balancing Domain
Decomposition by Constraints (BDDC) preconditioner that allows using the
conjugate gradient method to compute the solution of the saddle-point linear
systems arising from the VEM discretization of the three-dimensional Stokes
equations. We prove the scalability and quasi-optimality of the algorithm and
confirm the theoretical findings with parallel computations. Numerical results
with adaptively generated coarse spaces confirm the method's robustness in the
presence of large jumps in the viscosity and with high-order VEM
discretizations
Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations
Two novel parallel Newton-Krylov Balancing Domain Decomposition by
Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting
(FETI-DP) solvers are here constructed, analyzed and tested numerically for
implicit time discretizations of the three-dimensional Bidomain system of
equations.
This model represents the most advanced mathematical description of the
cardiac bioelectrical activity and it consists of a degenerate system of two
non-linear reaction-diffusion partial differential equations (PDEs), coupled
with a stiff system of ordinary differential equations (ODEs).
A finite element discretization in space and a segregated implicit
discretization in time, based on decoupling the PDEs from the ODEs, yields at
each time step the solution of a non-linear algebraic system.
The Jacobian linear system at each Newton iteration is solved by a Krylov
method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the
recently introduced {\em deluxe} scaling of the dual variables.
A polylogarithmic convergence rate bound is proven for the resulting parallel
Bidomain solvers.
Extensive numerical experiments on linux clusters up to two thousands
processors confirm the theoretical estimates, showing that the proposed
parallel solvers are scalable and quasi-optimal
Computational modeling of the electromechanical response of a ventricular fiber affected by eccentric hypertrophy
AbstractThe aim of this work is to study the effects of eccentric hypertrophy on the electromechanics of a single myocardial ventricular fiber by means of a one-dimensional finite-element strongly-coupled model. The electrical current ow model is written in the reference configuration and it is characterized by two geometric feedbacks, i.e. the conduction and convection ones, and by the mechanoelectric feedback due to stretchactivated channels. First, the influence of such feedbacks is investigated for both a healthy and a hypertrophic fiber in case of isometric simulations. No relevant discrepancies are found when disregarding one or more feedbacks for both fibers. Then, all feedbacks are taken into account while studying the electromechanical responses of fibers. The results from isometric tests do not point out any notable difference between the healthy and hypertrophic fibers as regards the action potential duration and conduction velocity. The length-tension relationships show increased stretches and reduced peak values for tension instead. The tension-velocity relationships derived from afterloaded isotonic and quick- release tests depict higher values of contraction velocity at smaller afterloads. Moreover, higher maximum shortenings are achieved during the isotonic contraction. In conclusion, our simulation results are innovative in predicting the electromechanical behavior of eccentric hypertrophic fibers
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