178 research outputs found
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
A full multigrid method for linear complementarity problems arising from elastic normal contact problems
This paper presents a full multigrid (FMG) technique, which combines
a multigrid method, an active set algorithm and a nested iteration technique, to solve
a linear complementarity problem (LCP) modeling elastic normal contact problems.
The governing system in this LCP is derived from a Fredholm integral of the rst
kind, and its coecient matrix is dense, symmetric and positive denite. One multigrid
cycle is applied to solve this system approximately in each active set iteration.
Moreover, this multigrid solver incorporates a special strategy to handle the complementarity
conditions, including restricting both the defect and the contact area
(active set) to the coarse grid, and setting all quantities outside contact to zero.
The smoother is chosen by some analysis based on the eigenvectors of the iteration
matrix. This method is applied to a Hertzian smooth contact and a rough surface
contact problem
The air gap between tape and drum in a video recorder
Lubrication with ambient air is not quite generally applied. The best known application is the "oil bearing" in tape recording systems for audio, video and computer applications; where the gap height that is needed for effective lubrication may quite easily be attained. This air gap reduces tape friction and wear of tape and recording head substantially. On the other hand, though, the air gap for effective magnetic recording is very small. These conflicting demands on the lubrication conditions ask for an accurate calculation of the air gap distribution.\ud
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The multigrid method — a fast, iterative equation solver — will be applied to calculate the film thickness in foil bearings on a fine grid. The results for a simple but adequate model for the air lubrication between the tape and the drum in a VHS-recorder will be presented, including details about the narrow gap between recording head and tape
A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction
International audienceIn this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth com-plementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples
A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit
In this work, we examine a numerical phase-field fracture framework in which
the crack irreversibility constraint is treated with a primal-dual active set
method and a linearization is used in the degradation function to enhance the
numerical stability. The first goal is to carefully derive from a
complementarity system our primal-dual active set formulation, which has been
used in the literature in numerous studies, but for phase-field fracture
without its detailed mathematical derivation yet. Based on the latter, we
formulate a modified combined active-set Newton approach that significantly
reduces the computational cost in comparison to comparable prior algorithms for
quasi-monolithic settings. For many practical problems, Newton converges fast,
but active set needs many iterations, for which three different efficiency
improvements are suggested in this paper. Afterwards, we design an iteration on
the linearization in order to iterate the problem to the monolithic limit. Our
new algorithms are implemented in the programming framework pfm-cracks [T.
Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field
fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the
numerical examples, we conduct performance studies and investigate efficiency
enhancements. The main emphasis is on the cost complexity by keeping the
accuracy of numerical solutions and goal functionals. Our algorithmic
suggestions are substantiated with the help of several benchmarks in two and
three spatial dimensions. Therein, predictor-corrector adaptivity and parallel
performance studies are explored as well.Comment: 49 pages, 45 figures, 9 table
On the contact detection for contact-impact analysis in multibody systems
One of the most important and complex parts of the simulation of multibody systems with contact-impact involves the detection of the precise instant of impact. In general, the periods of contact are very small and, therefore, the selection of the time step for the integration of the time derivatives of the state variables plays a crucial role in the dynamics of multibody systems. The conservative approach is to use very small time steps throughout the analysis. However, this solution is not efficient from the computational view point. When variable time step integration algorithms are used and the pre-impact dynamics does not involve high-frequencies the integration algorithms may use larger time steps and the contact between two surfaces may start with initial penetrations that are artificially high. This fact leads either to a stall of the integration algorithm or to contact forces that are physically impossible which, in turn, lead to post-impact dynamics that is unrelated to the physical problem. The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step, in variable time step integration algorithms, in the vicinity of contact of multibody systems. The proposed methodology ensures that for any impact in a multibody system the time step of the integration is such that any initial penetration is below any prescribed threshold. In the case of the start of contact, and after a time step is complete, the numerical error control of the selected integration algorithm is forced to handle the physical criteria to accept/reject time steps in equal terms with the numerical error control that it normally uses. The main features of this approach are the simplicity of its computational implementation, its good computational efficiency and its ability to deal with the transitions between non contact and contact situations in multibody dynamics. A demonstration case provides the results that support the discussion and show the validity of the proposed methodology.Fundação para a Ciência e a Tecnologia (FCT
Tikhonov-type iterative regularization methods for ill-posed inverse problems: theoretical aspects and applications
Ill-posed inverse problems arise in many fields of science and engineering. The ill-conditioning and the big dimension make the task of numerically solving this kind of problems very challenging.
In this thesis we construct several algorithms for solving ill-posed inverse problems. Starting from the classical Tikhonov regularization method we develop iterative methods that enhance the performances of the originating method.
In order to ensure the accuracy of the constructed algorithms we insert a priori knowledge on the exact solution and empower the regularization term. By exploiting the structure of the problem we are also able to achieve fast computation even when the size of the problem becomes very big.
We construct algorithms that enforce constraint on the reconstruction, like nonnegativity or flux conservation and exploit enhanced version of the Euclidian norm using a regularization operator and different semi-norms, like the Total Variaton, for the regularization term.
For most of the proposed algorithms we provide efficient strategies for the choice of the regularization parameters, which, most of the times, rely on the knowledge of the norm of the noise that corrupts the data.
For each method we analyze the theoretical properties in the finite dimensional case or in the more general case of Hilbert spaces.
Numerical examples prove the good performances of the algorithms proposed in term of both accuracy and efficiency
A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing
We present a parallel data structure which is directly linked to geometric quantities of an underlying mesh and which is well adapted to the requirements of a general finite element realization. In addition, we define an abstract linear algebra model which supports multigrid methods (extending our previous work in Comp. Vis. Sci. 1 (1997) 27-40). Finally, we apply the parallel multigrid preconditioner to several configurations in linear elasticity and we compute the condition number numerically for different smoothers, resulting in a quantitative evaluation of parallel multigrid performance
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