A Singularity-Free Penalty Formulation for the Dynamics of Constrained Multibody Systems

1992 ◽  
Author(s):  
A. Avello ◽  
E. Bayo
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Augmented Lagrangian ◽  
Jacobian Matrix ◽  
Augmented Lagrangian Method ◽  
Positive Definite Matrix ◽  
Dynamic Simulations ◽  
Classical Dynamic ◽  
Sudden Loss ◽  
Dynamic Formulation

Abstract In a singular position, the number of degrees of freedom of a mechanism instantaneously increases, which is detected by a sudden loss of rank in the jacobian matrix. This rank-deficiency causes the failure of the classical dynamic formulations. The enforcement of the constraints through a penalty method leads to an attractive and efficient dynamic formulation that, in addition, can be used at singularities. This formulation leads to a symmetric and positive definite matrix whose rank does not depend on the rank of the jacobian and thus is ideally suited for singular positions. A refinement of this approach is achieved through an augmented Lagrangian method. A simple example and two dynamic simulations show the effectiveness of the formulation.

An Efficient Method for the Kinematics of Multibody Systems That Works in Singular Positions

1992 ◽  
Author(s):  
A. Avello ◽  
E. Bayo
Keyword(s):  
Efficient Method ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Augmented Lagrangian ◽  
Multibody System ◽  
Jacobian Matrix ◽  
Numerical Examples ◽  
Constraint Equations ◽  
The Stability ◽  
Augmented Lagrangian Formulation

Abstract When a multibody system reaches a singular position, one or more degrees of freedom appear instantaneously and the jacobian matrix of the constraint equations becomes rank-deficient. The classical kinematic formulation is based on the factorization of the jacobian and, therefore, fails in singular positions. In this paper we develop an efficient method, which uses a penalty and an augmented Lagrangian formulation, and successfully handles singular positions. This formulation automatically copes with redundant incompatible constraints and guarantees the stability of the constraints during numerical integration. Critical numerical examples are shown which corroborate these findings.

Investigation of GPU Use in Conjunction With DCA-Based Articulated Multibody Systems Simulation

2015 ◽  
Author(s):  
Jeremy J. Laflin ◽  
Kurt S. Anderson ◽  
Michael Hans
Keyword(s):  
Multibody Systems ◽  
Graphics Processing Units ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Time ◽  
Test Case ◽  
Time Step ◽  
Dynamic Simulations ◽  
Computational Performance ◽  
Compute Time

Since computational performance is critically important for simulations to be used as an effective tool to study and design dynamic systems, the computing performance gains offered by Graphics Processing Units (GPUs) cannot be ignored. Since the GPU is designed to execute a very large number of simultaneous tasks (nominally Single Instruction Multi-Data (SIMD)), recursive algorithms in general, such as the DCA, are not well suited to be executed on GPU-type architecture. This is because each level of recursion is dependent on the previous level. However, there are some ways that the GPU can be leveraged to increase computational performance when using the DCA to form and solve the equations of motion for articulated multibody systems with a very large number of degrees-of-freedom. Computational performance of dynamic simulations is highly dependent on the nature of the underlying formulation and the number of generalized coordinates used to characterize the system. Therefore, algorithms that scale in a more desirable (lower order) fashion with the number of degrees-of-freedom are generally preferred when dealing with large (N > 10) systems. However, the utility of using simulations as a scientific tool is directly related to actual compute time. The DCA, and other top performing methods, have demonstrated the desirable property of the required compute time scaling linearly with (O(n)) with the number of degrees-of-freedom (n) and sublinearly (O(logn) performance when implemented in parallel. However for the DCA, total compute time could be further reduced by exploiting the large number of independent operations involved in the first few levels of recursion. A simple chain-type pendulum example is used to explore the feasibility of using the GPU to execute the assembly and disassembly operations for the levels of recursion that contain enough bodies for this process to be computationally advantageous. A multi-core CPU is used to perform the operations in parallel using Open MP for the remaining levels. The number of levels of recursion that utilizes the GPU is varied from zero to all levels. The data corresponding to zero utilization of the GPU provides the reference compute-time in which the assembly and disassembly operations necessary at each level are performed in parallel using Open MP. The computational time required to simulate the system for one time-step where the GPU is utilized for various levels of recursion is compared to the reference compute time also varying the number of bodies in the system. A decrease in the compute-time when using the GPU is demonstrated relative to the reference compute-time even for systems of moderate size n < 1000 for arrangements using the GPU. This is a lower number of bodies than was expected for this test case and confirms that the GPU can bring significant increases in computational efficiency for large systems, while preserving the attractive sub-linear scalability (w.r.t. compute time) of the DCA.

Application Criteria for Conserving Integrators and Projection Methods in Multibody Dynamics

2007 ◽  
Author(s):  
Daniel Dopico ◽  
Javier Cuadrado ◽  
Juan C. Garcia Orden ◽  
Alberto Luaces
Keyword(s):  
Recent Work ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Projection Methods ◽  
Orthogonal Projections ◽  
Dynamics Of Multibody Systems ◽  
Energy Conserving ◽  
Realistic Problem

This work presents the application to the dynamics of multibody systems of two methods based on augmented Lagrangian techniques, compares them, and gives some criteria for its use in realistic problems. The methods are an augmented Lagrangian method with orthogonal projections of velocities and accelerations, and an augmented Lagrangian energy conserving method. Both methods were presented by the authors in a very recent work, but it was not complete since the testing and the comparison of the methods was done by simulating a simple and academic example, and that was not sufficient to draw conclusions in terms of efficiency. For this work, the whole model of a vehicle has been simulated through both formulations, and their performance compared for such a large and realistic problem.

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

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