1,333 research outputs found
A robust adaptive algebraic multigrid linear solver for structural mechanics
The numerical simulation of structural mechanics applications via finite
elements usually requires the solution of large-size and ill-conditioned linear
systems, especially when accurate results are sought for derived variables
interpolated with lower order functions, like stress or deformation fields.
Such task represents the most time-consuming kernel in commercial simulators;
thus, it is of significant interest the development of robust and efficient
linear solvers for such applications. In this context, direct solvers, which
are based on LU factorization techniques, are often used due to their
robustness and easy setup; however, they can reach only superlinear complexity,
in the best case, thus, have limited applicability depending on the problem
size. On the other hand, iterative solvers based on algebraic multigrid (AMG)
preconditioners can reach up to linear complexity for sufficiently regular
problems but do not always converge and require more knowledge from the user
for an efficient setup. In this work, we present an adaptive AMG method
specifically designed to improve its usability and efficiency in the solution
of structural problems. We show numerical results for several practical
applications with millions of unknowns and compare our method with two
state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM
Parallel three-dimensional simulations of quasi-static elastoplastic solids
Hypo-elastoplasticity is a flexible framework for modeling the mechanics of
many hard materials under small elastic deformation and large plastic
deformation. Under typical loading rates, most laboratory tests of these
materials happen in the quasi-static limit, but there are few existing
numerical methods tailor-made for this physical regime. In this work, we extend
to three dimensions a recent projection method for simulating quasi-static
hypo-elastoplastic materials. The method is based on a mathematical
correspondence to the incompressible Navier-Stokes equations, where the
projection method of Chorin (1968) is an established numerical technique. We
develop and utilize a three-dimensional parallel geometric multigrid solver
employed to solve a linear system for the quasi-static projection. Our method
is tested through simulation of three-dimensional shear band nucleation and
growth, a precursor to failure in many materials. As an example system, we
employ a physical model of a bulk metallic glass based on the shear
transformation zone theory, but the method can be applied to any
elastoplasticity model. We consider several examples of three-dimensional shear
banding, and examine shear band formation in physically realistic materials
with heterogeneous initial conditions under both simple shear deformation and
boundary conditions inspired by friction welding.Comment: Final version. Accepted for publication in Computer Physics
Communication
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
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