Complementarity Techniques for Minimal Coordinate Contact Dynamics

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Harshavardhan Mylapilli ◽  
Abhinandan Jain
Keyword(s):  
Complementarity Problem ◽  
Optimization Problems ◽  
Nonlinear Complementarity Problem ◽  
Computational Cost ◽  
Contact Dynamics ◽  
Time Step ◽  
Frictional Contacts ◽  
Computational Costs ◽  
Simulation Results ◽  
Dynamics Problems

In this paper, nonsmooth contact dynamics of articulated rigid multibody systems is formulated as a complementarity problem. Minimal coordinate (MC) formulation is used to derive the dynamic equations of motion as it provides significant computational cost benefits and leads to a smaller-sized complementarity problem when compared with the frequently used redundant coordinate (RC) formulation. Additionally, an operational space (OS) formulation is employed to take advantage of the low-order structure-based recursive algorithms that do not require mass matrix inversion, leading to a further reduction in these computational costs. Based on the accuracy with which Coulomb's friction cone is modeled, the complementarity problem can be posed either as a linear complementarity problem (LCP), where the friction cone is approximated using a polygon, or as a nonlinear complementarity problem (NCP), where the friction cone is modeled exactly. Both formulations are studied in this paper. These complementarity problems are further recast as nonsmooth unconstrained optimization problems, which are solved by employing a class of Levenberg–Marquardt (LM) algorithms. The necessary theory detailing these techniques is discussed and five solvers are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these solvers for a single contact, whereas a 12-link complex pendulum example is chosen to compare the accuracy of the solvers for the case of multiple simultaneous contacts. The simulation results validate the MC-based NCP formulations developed in this paper. Moreover, we observe that the LCP solvers deliver accuracy comparable to that of the NCP solvers when the friction cone direction vectors in the contact tangent plane are aligned with the sliding contact velocity at each time step. The theory and simulation results show that the NCP approach can be seamlessly recast into an MC OS formulation, thus allowing for accurate modeling of frictional contacts, while at the same time reducing overall computational costs associated with contact and collision dynamics problems in articulated rigid body systems.

Evaluation of Complementarity Techniques for Minimal Coordinate Contact Dynamics

2014 ◽  
Author(s):  
Harshavardhan Mylapilli ◽  
Abhinandan Jain
Keyword(s):  
Complementarity Problem ◽  
Optimization Problems ◽  
Plane Surface ◽  
Equations Of Motion ◽  
Nonlinear Complementarity Problem ◽  
Contact Dynamics ◽  
Test Case ◽  
Coulomb’S Friction ◽  
Space Formulation ◽  
Multi Body

In this article, the non-smooth contact dynamics of multi-body systems is formulated as a complementarity problem. Minimal coordinates operational space formulation is used to derive the dynamics equations of motion. Depending on the approach used for modeling Coulomb’s friction, the complementarity problem can be posed either as a linear or a nonlinear problem. Both formulations are studied in this paper. An exact modeling of the friction cone leads to a nonlinear complementarity problem (NCP) formulation whereas a polyhedral approximation of the friction cone results in a linear complementarity problem (LCP) formulation. These complementarity problems are further recast as non-smooth unconstrained optimization problems, which are solved by employing a class of Levenberg-Marquardt algorithms. The necessary theory detailing these techniques is discussed and five schemes are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these schemes, while a twelve-link pendulum example is chosen to compare the speed and accuracy of the schemes presented in this paper.

scholarly journals Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

2019 ◽  
Vol 71 (5) ◽  
Author(s):  
Eduardo Corona ◽  
David Gorsich ◽  
Paramsothy Jayakumar ◽  
Shravan Veerapaneni
Keyword(s):  
Linear Systems ◽  
Complementarity Problem ◽  
Large Scale ◽  
Nonlinear Complementarity Problem ◽  
Second Order ◽  
Rigid Body Dynamics ◽  
Time Step ◽  
Newton Step ◽  
Primal Dual ◽  
Second Order Methods

Abstract In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred a renewed interest in the discrete element method (DEM) modeling of frictional contact. Force penalty methods, while economic and widely accessible, introduce artificial stiffness, requiring small time steps to retain numerical stability. Optimization-based methods, which enforce contacts geometrically through complementarity constraints leading to a differential variational inequality problem (DVI), allow for the use of larger time steps at the expense of solving a nonlinear complementarity problem (NCP) each time-step. We review the latest efforts to produce solvers for this NCP, focusing on its relaxation to a cone complementarity problem (CCP) and solution via an equivalent quadratic optimization problem with conic constraints. We distinguish between first-order methods, which use only gradient information and are thus linearly convergent and second-order methods, which rely on a Newton type step to gain quadratic convergence and are typically more robust and problem-independent. However, they require the approximate solution of large sparse linear systems, thus losing their competitive advantages in large scale problems due to computational cost. In this work, we propose a novel acceleration for the solution of Newton step linear systems in second-order methods using low-rank compression based fast direct solvers, leveraging on recent direct solver techniques for structured linear systems arising from differential and integral equations. We employ the quantized tensor train (QTT) decomposition to produce efficient approximate representations of the system matrix and its inverse. This provides a versatile and robust framework to accelerate its solution using this inverse in a direct or a preconditioned iterative method. We demonstrate compressibility of the Newton step matrices in primal dual interior point (PDIP) methods as applied to the multibody dynamics problem. Using a number of numerical tests, we demonstrate that this approach displays sublinear scaling of precomputation costs, may be efficiently updated across Newton iterations as well as across simulation time steps, and leads to a fast, optimal complexity solution of the Newton step. This allows our method to gain an order of magnitude speedups over state-of-the-art preconditioning techniques for moderate to large-scale systems, hence mitigating the computational bottleneck of second-order methods.

A NEURAL NETWORK FOR THE GENERALIZED NONLINEAR COMPLEMENTARITY PROBLEM OVER A POLYHEDRAL CONE

2015 ◽  
Vol 99 (3) ◽  
pp. 364-379 ◽  
Author(s):  
YIFEN KE ◽  
CHANGFENG MA
Keyword(s):  
Neural Network ◽  
Asymptotic Stability ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Unconstrained Minimization ◽  
Polyhedral Cone ◽  
Nonlinear Complementarity ◽  
Trajectory Simulation ◽  
The Neural Network ◽  
Simulation Results

In this paper, we consider a neural network model for solving the generalized nonlinear complementarity problem (denoted by GNCP) over a polyhedral cone. The neural network is derived from an equivalent unconstrained minimization reformulation of the GNCP, which is based on the penalized Fischer–Burmeister function ${\it\phi}_{{\it\mu}}(a,b)={\it\mu}{\it\phi}_{\mathit{FB}}(a,b)+(1-{\it\mu})a_{+}b_{+}$. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability and exponential stability. It is found that a larger ${\it\mu}$ leads to a better convergence rate of the trajectory. Simulation results are also reported.

Formulation of Multibody Dynamics as Complementarity Problems

2003 ◽  
Author(s):  
J. C. Trinkle
Keyword(s):  
Discrete Time ◽  
Complementarity Problem ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Contact Forces ◽  
Integration Scheme ◽  
Time Step ◽  
Stick Slip ◽  
Frictional Contacts ◽  
Time Formulation

Multibody systems with rigid bodies and unilateral contacts are difficult to simulate due to discontinuities associated with gaining and losing contacts and stick-slip transitions. Methods for simulating such systems fall into two categories: penalty methods and complementarity methods. The former calculate penetration depths of virtual rigid bodies at every time step and compute restoring forces to repair penetrations, while the latter assume that the bodies are truly rigid and compute contact forces that prevent penetration from occurring at all. In this paper, we are concerned with complementarity methods. We present an instantaneous formulation of the equations of motion of multi-rigid-body systems with frictional contacts as a complementarity problem. The unknowns in this formulation are accelerations and forces at the contacts. Since it is known that this model does not always admit a finite solution, it is problematic to use it directly in an integration scheme. This fact motivates the discrete-time formulation presented second. Although the discrete-time formulation also takes the form of a complementarity problem, it does not suffer from non-existence, and thus it is suitable for simulation. Numerical results are compared to the exact solution for a sphere initially sliding, then rolling, on a horizontal plane.

scholarly journals Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Atmosphere ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 444 ◽  
Author(s):  
Jinxi Li ◽  
Jie Zheng ◽  
Jiang Zhu ◽  
Fangxin Fang ◽  
Christopher. Pain ◽  
...  
Keyword(s):  
Unstructured Mesh ◽  
Computational Cost ◽  
Adaptive Mesh ◽  
Galerkin Finite Element Method ◽  
Adaptive Meshes ◽  
Galerkin Finite Element ◽  
Computational Costs ◽  
Cut Cell ◽  
Discontinuous Galerkin Finite Element ◽  
Terrain Following

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

scholarly journals Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework

2021 ◽  
Author(s):  
Tarun Gangwar ◽  
Dominik Schillinger
Keyword(s):  
Optimization Problems ◽  
Computational Cost ◽  
Structure Optimization ◽  
Material Behavior ◽  
Hierarchical Systems ◽  
Material Optimization ◽  
Continuum Micromechanics ◽  
Benchmark Tests ◽  
Constraint Optimization Problems ◽  
First Time

AbstractWe present a concurrent material and structure optimization framework for multiphase hierarchical systems that relies on homogenization estimates based on continuum micromechanics to account for material behavior across many different length scales. We show that the analytical nature of these estimates enables material optimization via a series of inexpensive “discretization-free” constraint optimization problems whose computational cost is independent of the number of hierarchical scales involved. To illustrate the strength of this unique property, we define new benchmark tests with several material scales that for the first time become computationally feasible via our framework. We also outline its potential in engineering applications by reproducing self-optimizing mechanisms in the natural hierarchical system of bamboo culm tissue.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

scholarly journals Optimization of injection molding process for car fender in consideration of energy efficiency and product quality

2014 ◽  
Vol 1 (4) ◽  
pp. 256-265 ◽  
Author(s):  
Hong Seok Park ◽  
Trung Thanh Nguyen
Keyword(s):  
Energy Efficiency ◽  
Energy Consumption ◽  
Injection Molding ◽  
Product Quality ◽  
Process Parameters ◽  
Optimization Problems ◽  
Computational Cost ◽  
Injection Molding Process ◽  
Molding Process ◽  
Nsga Ii

Abstract Energy efficiency is an essential consideration in sustainable manufacturing. This study presents the car fender-based injection molding process optimization that aims to resolve the trade-off between energy consumption and product quality at the same time in which process parameters are optimized variables. The process is specially optimized by applying response surface methodology and using nondominated sorting genetic algorithm II (NSGA II) in order to resolve multi-object optimization problems. To reduce computational cost and time in the problem-solving procedure, the combination of CAE-integration tools is employed. Based on the Pareto diagram, an appropriate solution is derived out to obtain optimal parameters. The optimization results show that the proposed approach can help effectively engineers in identifying optimal process parameters and achieving competitive advantages of energy consumption and product quality. In addition, the engineering analysis that can be employed to conduct holistic optimization of the injection molding process in order to increase energy efficiency and product quality was also mentioned in this paper.

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