71,957 research outputs found
Automatic Frechet differentiation for the numerical solution of boundary-value problems
A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Frechet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Frechet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space
A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation
Compressible Mooney-Rivlin theory has been used to model hyperelastic solids,
such as rubber and porous polymers, and more recently for the modeling of soft
tissues for biomedical tissues, undergoing large elastic deformations. We
propose a solution procedure for Lagrangian finite element discretization of a
static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the
case in which the boundary condition is a large prescribed deformation, so that
mesh tangling becomes an obstacle for straightforward algorithms. Our solution
procedure involves a largely geometric procedure to untangle the mesh: solution
of a sequence of linear systems to obtain initial guesses for interior nodal
positions for which no element is inverted. After the mesh is untangled, we
take Newton iterations to converge to a mechanical equilibrium. The Newton
iterations are safeguarded by a line search similar to one used in
optimization. Our computational results indicate that the algorithm is up to 70
times faster than a straightforward Newton continuation procedure and is also
more robust (i.e., able to tolerate much larger deformations). For a few
extremely large deformations, the deformed mesh could only be computed through
the use of an expensive Newton continuation method while using a tight
convergence tolerance and taking very small steps.Comment: Revision of earlier version of paper. Submitted for publication in
Engineering with Computers on 9 September 2010. Accepted for publication on
20 May 2011. Published online 11 June 2011. The final publication is
available at http://www.springerlink.co
A global method for coupling transport with chemistry in heterogeneous porous media
Modeling reactive transport in porous media, using a local chemical
equilibrium assumption, leads to a system of advection-diffusion PDE's coupled
with algebraic equations. When solving this coupled system, the algebraic
equations have to be solved at each grid point for each chemical species and at
each time step. This leads to a coupled non-linear system. In this paper a
global solution approach that enables to keep the software codes for transport
and chemistry distinct is proposed. The method applies the Newton-Krylov
framework to the formulation for reactive transport used in operator splitting.
The method is formulated in terms of total mobile and total fixed
concentrations and uses the chemical solver as a black box, as it only requires
that on be able to solve chemical equilibrium problems (and compute
derivatives), without having to know the solution method. An additional
advantage of the Newton-Krylov method is that the Jacobian is only needed as an
operator in a Jacobian matrix times vector product. The proposed method is
tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009)
http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1
Preconditioned fully implicit PDE solvers for monument conservation
Mathematical models for the description, in a quantitative way, of the
damages induced on the monuments by the action of specific pollutants are often
systems of nonlinear, possibly degenerate, parabolic equations. Although some
the asymptotic properties of the solutions are known, for a short window of
time, one needs a numerical approximation scheme in order to have a
quantitative forecast at any time of interest. In this paper a fully implicit
numerical method is proposed, analyzed and numerically tested for parabolic
equations of porous media type and on a systems of two PDEs that models the
sulfation of marble in monuments. Due to the nonlinear nature of the underlying
mathematical model, the use of a fixed point scheme is required and every step
implies the solution of large, locally structured, linear systems. A special
effort is devoted to the spectral analysis of the relevant matrices and to the
design of appropriate iterative or multi-iterative solvers, with special
attention to preconditioned Krylov methods and to multigrid procedures.
Numerical experiments for the validation of the analysis complement this
contribution.Comment: 26 pages, 13 figure
Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems
We present a method for approximating the solution of a parameterized,
nonlinear dynamical system using an affine combination of solutions computed at
other points in the input parameter space. The coefficients of the affine
combination are computed with a nonlinear least squares procedure that
minimizes the residual of the governing equations. The approximation properties
of this residual minimizing scheme are comparable to existing reduced basis and
POD-Galerkin model reduction methods, but its implementation requires only
independent evaluations of the nonlinear forcing function. It is particularly
appropriate when one wishes to approximate the states at a few points in time
without time marching from the initial conditions. We prove some interesting
characteristics of the scheme including an interpolatory property, and we
present heuristics for mitigating the effects of the ill-conditioning and
reducing the overall cost of the method. We apply the method to representative
numerical examples from kinetics - a three state system with one parameter
controlling the stiffness - and conductive heat transfer - a nonlinear
parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table
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