Penalty Method (penalty + method)

Distribution by Scientific Domains


Selected Abstracts


Some numerical issues using element-free Galerkin mesh-less method for coupled hydro-mechanical problems

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2009
Mohammad Norouz Oliaei
Abstract A new formulation of the element-free Galerkin (EFG) method is developed for solving coupled hydro-mechanical problems. The numerical approach is based on solving the two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Spatial variables in the weak form, i.e. displacement increment and pore water pressure increment, are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on a penalty method. Numerical stability of the developed formulation is examined in order to achieve appropriate accuracy of the EFG solution for coupled hydro-mechanical problems. Examples are studied and compared with closed-form or finite element method solutions to demonstrate the validity of the developed model and its capabilities. The results indicate that the EFG method is capable of handling coupled problems in saturated porous media and can predict well both the soil deformation and variation of pore water pressure over time. Some guidelines are proposed to guarantee the accuracy of the EFG solution for coupled hydro-mechanical problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]


Imposition of essential boundary conditions by displacement constraint equations in meshless methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2001
Xiong Zhang
Abstract One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary ,u, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov,Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd. [source]


A modified node-to-segment algorithm passing the contact patch test

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Giorgio Zavarise
Abstract Several investigations have shown that the classical one-pass node-to-segment (NTS) algorithms for the enforcement of contact constraints fail the contact patch test. This implies that the algorithms may introduce solution errors at the contacting surfaces, and these errors do not necessarily decrease with mesh refinement. The previous research has mainly focused on the Lagrange multiplier method to exactly enforce the contact geometry conditions. The situation is even worse with the penalty method, due to its inherent approximation that yields a solution affected by a non-zero penetration. The aim of this study is to analyze and improve the contact patch test behavior of the one-pass NTS algorithm used in conjunction with the penalty method for 2D frictionless contact. The paper deals with the case of linear elements. For this purpose, several sequential modifications of the basic formulation have been considered, which yield incremental improvements in results of the contact patch test. The final proposed formulation is a modified one-pass NTS algorithm which is able to pass the contact patch test also if used in conjunction with the penalty method. In other words, this algorithm is able to correctly reproduce the transfer of a constant contact pressure with a constant proportional penetration. Copyright © 2009 John Wiley & Sons, Ltd. [source]


A contact algorithm for frictional crack propagation with the extended finite element method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2008
Fushen Liu
Abstract We present an incremental quasi-static contact algorithm for path-dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non-linear contact mechanics and embedded within a localized element by penalty method. Newton,Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ,LATIN' for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd. [source]


Symmetry preserving algorithm for large displacement frictionless contact by the pre-discretization penalty method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
D. Gabriel
Abstract A three-dimensional contact algorithm based on the pre-discretization penalty method is presented. Using the pre-discretization formulation gives rise to contact searching performed at the surface Gaussian integration points. It is shown that the proposed method is consistent with the continuum formulation of the problem and allows an easy incorporation of higher-order elements with midside nodes to the analysis. Moreover, a symmetric treatment of mutually contacting surfaces is preserved even under large displacement increments. The proposed algorithm utilizes the BFGS method modified for constrained non-linear systems. The effectiveness of quadratic isoparametric elements in contact analysis is tested in terms of numerical examples verified by analytical solutions and experimental measurements. The symmetry of the algorithm is clearly manifested in the problem of impact of two elastic cylinders. Copyright © 2004 John Wiley & Sons, Ltd. [source]


Die shape design optimization of sheet metal stamping process using meshfree method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2001
Nam Ho Kim
Abstract A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity-based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum-based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation-dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring-back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd. [source]


Weak imposition of boundary conditions for the Navier,Stokes equations by a penalty method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2009
Atife Caglar
Abstract We prove convergence of the finite element method for the Navier,Stokes equations in which the no-slip condition and no-penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier,Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd. [source]


Hyperbolic Penalty: A New Method for Nonlinear Programming with Inequalities

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 6 2001
Adilson Elias Xavier
This work intends to present and to analyze a new penalty method that purposes to solve the general nonlinear programming problem subject to inequality constraints. The proposed method has the important feature of being completely differentiable and combines features of both exterior and interior penalty methods. Numerical results for some problems are commented on. [source]


Multigrid methods for the symmetric interior penalty method on graded meshes

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2009
S. C. Brenner
Abstract The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd. [source]


Using logarithmic penalties in the shooting algorithm for optimal control problems

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 5 2003
J. F. Bonnans
Abstract The paper deals with optimal control problems of ordinary differential equations with bound control constraints. We analyse the logarithmic penalty method for converting the problem into an unconstrained one, the latter being solved by a shooting algorithm. Convergence of the value function and optimal controls is obtained for linear quadratic problems, and more generally when the control variable enters linearly in the state equation and in a quadratic way in the cost function. We display some numerical results on two examples: an aircraft maneuver, and the stabilization of an oscillating system. Copyright © 2003 John Wiley & Sons, Ltd. [source]


On an implicit particle method for simulation of forming processes

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005
O. Schilling
Task is the simulation of forming processes using particle methods. We implemented some mesh-free methods (the element free Galerkin method [1] and others) and the finite element method in one programme system which permits a direct comparison. For the mesh-free methods a moving least squares approximation is applied. The shape functions are not zero or one at the nodes, thus essential boundary conditions cannot be imposed directly [2]. We use a penalty method to enforce essential boundary conditions and contact conditions. The contact algorithm (normal contact of nodes to C1 -continuous surfaces) is checked by means of the element free Galerkin method and the FEM on the basis of numerical examples which deal with forming processes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Alternative Definitions of Micro,to,Macro Transitions in Particle Aggregates of Granular Materials

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Joachim Dettmar Dipl.-Ing.
An alternative approach to the modelling of the overall macroscopic response of granular materials is presented based on micro,macro transitions. A homogenized macro,continuum is considered with locally attached granular microstructures. The new developments include a consistent transfer of micro,macro transitions from continuous to particle microstructures with a new overall macro,stress definition. Secondly, a new class of boundary conditions for displacements/ rotations on prescribed boundary,frame particles is presented that yields upper and lower bounds with regard to the aggregate stiffness. Thirdly, a unified implementation of the displacement/ rotation boundary constraints by a penalty method is developed that proves to be a convenient computational tool in explicit codes. [source]


Advances in collision detection and non-linear finite mixed element modelling for improved soft tissue simulation in craniomaxillofacial surgical planning

THE INTERNATIONAL JOURNAL OF MEDICAL ROBOTICS AND COMPUTER ASSISTED SURGERY, Issue 1 2010
Shengzheng Wang
Abstract Background There is a huge demand to develop a method for assisting surgeons in automatically predicting soft tissue deformation in terms of a bone-remodelling plan. Methods This paper introduces several novel elements into a system for the simulation of postoperative facial appearances with respect to prespecified bone-remodelling plans. First, a new algorithm for efficient detection of collisions, using the signed distance field, is described. Next, the penalty method is applied to determine the contact load of bone on facial soft tissue. Finally, a non-linear finite mixed element model is developed to estimate the tissue deformation induced by the prescribed bone remodelling plan. Results The performance of the proposed collision detection algorithm has been improved in memory requirements and computational efficiency compared with conventional methods. In addition, the methodology is evaluated over both synthetic and real data, with simulation performance averaging <0.5 mm pointwise error over the facial surface in six mid-face distraction osteotogenesis procedures. Conclusions The experimental results support the novel methodological advancements in collision detection and biomechanical modelling proposed in this work. Copyright © 2009 John Wiley & Sons, Ltd. [source]