Extended Divide-and-Conquer Algorithm for Uncertainty Analysis of Multibody Systems in Polynomial Chaos Expansion Framework

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Mohammad Poursina
Keyword(s):  
Computational Complexity ◽  
Polynomial Chaos ◽  
Equations Of Motion ◽  
Divide And Conquer ◽  
Chaos Expansion ◽  
Entire System ◽  
Governing Equations ◽  
Divide And Conquer Algorithm ◽  
Base Functions ◽  
Modal Values

In this paper, an advanced algorithm is presented to efficiently form and solve the equations of motion of multibody problems involving uncertainty in the system parameters and/or excitations. Uncertainty is introduced to the system through the application of polynomial chaos expansion (PCE). In this scheme, states of the system, nondeterministic parameters, and constraint loads are projected onto the space of specific orthogonal base functions through modal values. Computational complexity of traditional methods of forming and solving the resulting governing equations drastically grows as a cubic function of the size of the nondeterministic system which is significantly larger than the original deterministic multibody problem. In this paper, the divide-and-conquer algorithm (DCA) will be extended to form and solve the nondeterministic governing equations of motion avoiding the construction of the mass and Jacobian matrices of the entire system. In this strategy, stochastic governing equations of motion of each individual body as well as those associated with kinematic constraints at connecting joints are developed in terms of base functions and modal terms. Then using the divide-and-conquer scheme, the entire system is swept in the assembly and disassembly passes to recursively form and solve nondeterministic equations of motion for modal values of spatial accelerations and constraint loads. In serial and parallel implementations, computational complexity of the method is expected to, respectively, increase as a linear and logarithmic function of the size.

An Efficient Application of Polynomial Chaos Expansion for the Dynamic Analysis of Multibody Systems With Uncertainty

2014 ◽  
Author(s):  
Mohammad Poursina
Keyword(s):  
Computational Complexity ◽  
Maximum Degree ◽  
Polynomial Chaos ◽  
Multibody System ◽  
Equations Of Motion ◽  
Polynomial Chaos Expansion ◽  
Stochastic Equations ◽  
Basis Functions ◽  
Chaos Expansion ◽  
Modal Values

In this paper the mathematical framework of an advanced algorithm is presented to efficiently form and solve the equations of motion of a multibody system involving uncertainty in the system parameters and/or the excitations. The uncertainty is introduced to the system through the application of the polynomial chaos expansion. In this scheme, states of the system, nondeterministic parameters, and the constraint loads are expanded using modal values as well as orthogonal basis functions. Computational complexity of the application of traditional methods to solve the stochastic equations of motion of a multibody system drastically grows as a cubic function of the number of the states of the system, uncertain parameters and the maximum degree of the polynomial chosen for the basis function. The presented method forms the equation of motion of the system without forming the entire mass and Jacobian matrices. In this strategy, the stochastic governing equations of motion of each individual body as well as the one associated with the kinematic constraint at the connecting joint are developed in terms of the basis functions and modal coordinates. Then sweeping the system in two passes assembly and disassembly, one can form and solve the stochastic equations of motion. In the assembly pass the non-deterministic equations of motion of the assemblies are obtained. In the disassembly process, these equations are then recursively solved for the modal values of the spatial accelerations and the constraints loads. In the serial and parallel implementations, computational complexity of the method increases as a linear and logarithmic functions of the number of the states of the system, uncertain variables, and the maximum degree of the basis functions used in the expansion.

The Rapid Uncertainty Prediction of the Ocean-Acoustic Coupled Model

2020 ◽  
Author(s):  
Baolong Cui ◽  
Wuhong Guo
Keyword(s):  
Computational Complexity ◽  
Acoustic Field ◽  
Sound Speed ◽  
Polynomial Chaos ◽  
Coupled Model ◽  
Polynomial Chaos Expansion ◽  
Chaos Expansion ◽  
Uncertainty Distribution ◽  
Uncertainty Prediction ◽  
Sound Speed Structure

<p>Focusing on the rapid prediction of acoustic field uncertainty in environment with temporal and spatial sound speed perturbation, evolvement of sound speed structure over time is predicted based on the ocean-acoustic coupled model to obtain the uncertainty distribution of the vertical structure of sound speed. Further, a method combining  the arbitrary polynomial chaos expansion with the empirical orthogonal function is proposed to reduce the dimensionality of uncertain parameters and to obtain the uncertainty distribution of the acoustic field. Simulations have shown that the computational complexity can be reduced by 2 orders of magnitude compared to the conventional polynomial chaos expansion while ensures the same precision. Moreover, the computational complexity is not influenced by the complexity of the sound speed profile. The acoustic field and uncertainty predicted in uncertain environment by proposed method also have been tested with the experimental data.</p>

Multi-Flexible-Body Simulations Using Interpolating Splines in a Divide-and-Conquer Scheme

2013 ◽  
Author(s):  
Imad M. Khan ◽  
Woojin Ahn ◽  
Kurt Anderson ◽  
Suvranu De
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Divide And Conquer Algorithm ◽  
Large Rotations ◽  
Floating Frame ◽  
Interpolating Splines ◽  
Divide And Conquer Scheme ◽  
Logarithmic Complexity

A new method for modeling multi-flexible-body systems is presented that incorporates interpolating splines in a divide-and-conquer scheme. This algorithm uses the floating frame of reference formulation and piece-wise interpolation spline functions to construct and solve the non-linear equations of motion of the multi-flexible-body systems undergoing large rotations and translations. We compare the new algorithm with the flexible divide-and-conquer algorithm (FDCA) that uses the assumed modes method and may resort to sub-structuring in many cases [1]. We demonstrate, through numerical examples, that in such cases the interpolating spline-based approach is comparable in accuracy and superior in efficiency to the FDCA. The algorithm retains the theoretical logarithmic complexity inherent to the divide-and-conquer algorithm when implemented in parallel.

Forward Kinematic Analysis of Non-Deterministic Articulated Multibody Systems With Kinematically Closed-Loops in Polynomial Chaos Expansion Scheme

2015 ◽  
Author(s):  
Sahand Sabet ◽  
Mohammad Poursina
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Kinematic Analysis ◽  
Multibody Systems ◽  
Polynomial Chaos ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Closed Loops ◽  
Base Functions ◽  
Forward Kinematic

This paper presents the method of polynomial chaos expansion (PCE) for the forward kinematic analysis of nondeterministic multibody systems with kinematically closed-loops. The PCE provides an efficient mathematical framework to introduce uncertainty to the system. This is accomplished by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial base functions. This paper presents the detailed formulation of the kinematics of a constrained multibody system at the position, velocity, and acceleration levels in the PCE scheme. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Furthermore, forward kinematic analysis is conducted at the position, velocity, and acceleration levels for a non-deterministic four-bar mechanism with single and multiple uncertain parameters in the length of linkages of the system. Time efficiency and accuracy of the intrusive PCE approach are compared with the traditionally used Monte Carlo method. The results demonstrate the drastic increase in the computational time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters while the intrusive PCE provides better accuracy with much less computation complexity.

Advances in the Application of the Divide-and-Conquer Algorithm to Multibody System Dynamics

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Jeremy J. Laflin ◽  
Kurt S. Anderson ◽  
Imad M. Khan ◽  
Mohammad Poursina
Keyword(s):  
Computational Complexity ◽  
Task Scheduling ◽  
Constrained Systems ◽  
Divide And Conquer ◽  
Mathematical Framework ◽  
Complete Understanding ◽  
Basic Algorithm ◽  
Flexible Bodies ◽  
Divide And Conquer Algorithm ◽  
Serial Task

This paper summarizes the various recent advancements achieved by utilizing the divide-and-conquer algorithm (DCA) to reduce the computational burden associated with many aspects of modeling, designing, and simulating articulated multibody systems. This basic algorithm provides a framework to realize O(n) computational complexity for serial task scheduling. Furthermore, the framework of this algorithm easily accommodates parallel task scheduling, which results in coarse-grain O(log n) computational complexity. This is a significant increase in efficiency over forming and solving the Newton–Euler equations directly. A survey of the notable previous work accomplished, though not all inclusive, is provided to give a more complete understanding of how this algorithm has been used in this context. These advances include applying the DCA to constrained systems, flexible bodies, sensitivity analysis, contact, and hybridization with other methods. This work reproduces the basic mathematical framework for applying the DCA in each of these applications. The reader is referred to the original work for the details of the discussed methods.

Dynamics and Control of Multi-Flexible-Body Systems in a Divide-and-Conquer Scheme

2015 ◽  
Author(s):  
Imad M. Khan ◽  
Kalyan C. Addepalli ◽  
Mohammad Poursina
Keyword(s):  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Divide And Conquer ◽  
Flexible Body ◽  
Constraint Forces ◽  
Control Laws ◽  
Motion Constraints ◽  
Divide And Conquer Algorithm ◽  
Dynamics And Control ◽  
Kinematic Loops

In this paper, we present an extension of the generalized divide-and-conquer algorithm (GDCA) for modeling constrained multi-flexible-body systems. The constraints of interest in this case are not the motion constraints or the presence of closed kinematic loops but those that arise due to inverse dynamics or control laws. The introductory GDCA paper introduced an efficient methodology to include generalized constraint forces in the handle equations of motion of the original divide-and-conquer algorithm for rigid multibody systems. Here, the methodology is applied to flexible bodies connected by kinematic joints. We develop necessary equations that define the algorithm and present a well known numerical example to validate the method.

scholarly journals Tackling random fields non-linearities with unsupervised clustering of polynomial chaos expansion in latent space: application to global sensitivity analysis of river flooding

2021 ◽  
Author(s):  
Siham El Garroussi ◽  
Sophie Ricci ◽  
Matthias De Lozzo ◽  
Nicole Goutal ◽  
Didier Lucor
Keyword(s):  
Sensitivity Analysis ◽  
Water Depth ◽  
Polynomial Chaos ◽  
Global Sensitivity Analysis ◽  
Polynomial Chaos Expansion ◽  
Divide And Conquer ◽  
Chaos Expansion ◽  
Global Sensitivity ◽  
Numerical Solver ◽  
The Impact

AbstractA surrogate model is developed to accurately approximate a two-dimensional hydrodynamics numerical solver in order to conduct a reduced-cost variance-based global sensitivity analysis of the hydraulic state. The impact of uncertainties in river bottom friction and boundary conditions on the simulated water depth is analyzed for quasi-unsteady flows. An autoencoder technique adapted to non-linear variable dimension reduction is used to reduce the multi-dimensional model output so that the formulation of the surrogate remains computationally parsimonious. In addition, following the divide-and-conquer principle, a mixture of local polynomial chaos expansions is proposed to deal with non-linearity in the hydraulic state with respect to uncertain inputs. Machine learning techniques are used to automatically partition the input space into clusters that are not affected by non-linearities and support accurate surrogates. This combined strategy is applied to a reach of the Garonne River where river and floodplains dynamics are simulated by the numerical solver Telemac-2D. The merits of this strategy are highlighted when the flood front reaches regions where the topography features a strong gradient and where, consequently, strong non-linearities occur between the water depth and friction as well as hydrologic input forcing. By applying this strategy, the $$Q_2$$ Q 2 metric improves by 90% compared to a classical polynomial chaos expansion surrogate, resulting in a much more reliable sensitivity analysis. This is particularly important in floodplain areas where human and economic activities are at stake.

Computed Torque Control of Articulated Multibody Systems in the Generalized Divide and Conquer Algorithm Framework

2015 ◽  
Author(s):  
Cameron Kingsley ◽  
Mohammad Poursina
Keyword(s):  
Multibody Systems ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Torque Control ◽  
Divide And Conquer ◽  
Controlled System ◽  
Divide And Conquer Algorithm ◽  
Computed Torque Control ◽  
And Control ◽  
Computed Torque

An extension to the Generalized-Divide-and-Conquer Algorithm (GDCA) is presented in this paper in conjunction with the Computed-Torque-Control-Law (CTCL) to model and control fully actuated multibody systems. CTCL uses the inverse dynamics to provide control inputs to the system while, the dynamics of the system must be formed and solved in each iteration. Herein, the GDCA is extended to form and solve the inverse dynamics to find control torques. Further, this method is also extended to efficiently solve the equations of motion of the controlled system. This significantly reduces the complexity of modeling, simulating, and controlling the fully actuated multibody systems to O(n) or O(logn) operations in each iteration in the serial and parallel implementations, respectively.

Generalized Orthogonal Complement Based Divide and Conquer Algorithm for Constrained Multibody Dynamics

2009 ◽  
Author(s):  
Rudranarayan M. Mukherjee ◽  
Kurt S. Anderson
Keyword(s):  
Equations Of Motion ◽  
Divide And Conquer ◽  
Orthogonal Complement ◽  
Assembly Process ◽  
Short Article ◽  
Current Form ◽  
Divide And Conquer Algorithm ◽  
Constrained Multibody Dynamics ◽  
Kinematic Loops ◽  
Generalized Orthogonal

This paper presents an extension of the orthogonal complement based divide and conquer algorithm for constraint multi-rigid body systems containing closed kinematic loops in generalized topologies. In its current form, its a short article demonstrating the methodology for assembling the equations of motion in a hierarchic assembly process for systems containing multiple loops in generalized topologies.

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