Relevant bibliographies by topics / Jacobian matrix H

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Academic literature on the topic 'Jacobian matrix H'

Author: Grafiati
Published: 3 June 2025
Last updated: 11 July 2025

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Journal articles on the topic "Jacobian matrix H"

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Alwardat, Mohammad Y. "GEOMETRIC JACOBIANS DERIVATION AND KINEMATIC SINGULARITY ANALYSIS FOR 6-DOF ROBOTIC MANIPULATOR." international journal of advanced research in computer science 16, no. 1 (2025): 6–20. https://doi.org/10.26483/ijarcs.v16i1.7178.

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This article investigates the kinematic singularities and geometric Jacobians of a 6-DOF robotic manipulator, incorporating a prismatic joint, from the perspective of singularity theory. The study begins by deriving the forward kinematics using the Denavit-Hartenberg (D-H) convention and examines the Jacobian matrices to identify configurations where the Jacobian matrix becomes rank-deficient, signaling the presence of kinematic singularities. These singularities pose critical challenges, such as restricting end-effector mobility and leading to infinite solutions in inverse kinematics. The determinant of the Jacobian matrix is employed to detect singular configurations, and the implications for motion control and trajectory planning are discussed. Through a detailed analysis and MATLAB simulations, the article highlights the importance of singularity avoidance and provides a deeper understanding of the manipulator's kinematic behavior. The findings emphasize the need for strategic design and motion planning to ensure optimal performance and stability in robotic manipulation tasks
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Efimov, Danil, Anatoliy Petravchuk, and Maksym Sydorov. "Centralizers of Jacobian derivations." Algebra and Discrete Mathematics 36, no. 1 (2023): 22–31. http://dx.doi.org/10.12958/adm2186.

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Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.
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Butuk, N., and J. P. Pemba. "Computing CHEMKIN Sensitivities Using Complex Variables." Journal of Engineering for Gas Turbines and Power 125, no. 3 (2003): 854–58. http://dx.doi.org/10.1115/1.1469006.

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This paper discusses an accurate numerical approach based on complex variables for the computation of the Jacobian matrix of complex chemical reaction mechanisms. The Jacobian matrix is required in the calculation of low dimensional manifolds during kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large-scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a nonlinear reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size h.
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Yan, Yu, and Gong Ping Wu. "The Jacobian Matrix Structure of the Barrier Arm of Inspection Robot." Advanced Materials Research 706-708 (June 2013): 1183–86. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.1183.

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The Jacobian matrix represents the relationship of the linear mapping between the space velocity of robot operation and joint space velocity, which is the important link in the process of robot control. According to analyzing the gesture character of the inspection robot when it gets over the obstacles. This paper sets up the dynamitic model of robot by utilizing D-H. Based on that, it builds the Jacobian matrix by adopting differential transformation method, which establishes the foundation of the robots motion planning and real-time control.
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Rong, Xue Wen, Yi Bin Li, Jiu Hong Ruan, and Hong Jun Song. "Kinematics Analysis and Simulation of a Quadruped Robot." Applied Mechanics and Materials 26-28 (June 2010): 517–22. http://dx.doi.org/10.4028/www.scientific.net/amm.26-28.517.

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This paper presents the mechanical configuration of a quadruped robot with 4 rotary joints per leg firstly. All joints of the robot are activated by 16 identical hydraulic servo cylinders. Then it gives the forward and inverse kinematic equations of one leg with D-H transformation matrix. Furthermore, it also gives the jacobian matrix of the legs. The jacobian matrix establishes the relationships between the angle velocities of the 4 joints of the supporting leg and the moving velocity of the robot trunk. Lastly, a simple trot gait simulation is given based on MSC.ADAMS under the assumption that there are no slippages between the supporting feet and the ground.
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Yang, Fang, Lihui Yang, and Xikui Ma. "Fast-Scale Bifurcation Analysis in One-Cycle Controlled H-Bridge Inverter." International Journal of Bifurcation and Chaos 26, no. 12 (2016): 1650199. http://dx.doi.org/10.1142/s0218127416501996.

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This paper discusses the fast-scale instability in the one-cycle controlled H-bridge inverter which is widely used in practical applications. Simulations show that the fast-scale instability which occurs around the peak value of the inductive current is a type of local instability. The discrete-time model is derived to analyze the fast-scale instability. According to the Jacobian matrix derived from the discrete-time model, the theoretical analysis shows that the fast-scale instability manifests itself as a period-doubling bifurcation occurs, which reveals the intrinsic mechanism of the instability of one-cycle controlled H-bridge inverter. Furthermore, with the approximation of the matrix exponential in the discrete-time model and based on the Jacobian matrix, the stability boundary is obtained by the analytical expression which is helpful to select the proper values of parameters for avoiding the fast-scale instability. The experiments are carried out to validate the results of the simulation and theoretical analysis. It is shown that the analysis method is able to facilitate the design of the inverter.
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DE BONDT, MICHIEL, and XIAOSONG SUN. "CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO." Bulletin of the Australian Mathematical Society 98, no. 1 (2018): 89–101. http://dx.doi.org/10.1017/s0004972718000345.

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Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.
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Lu, Jian, Guan Bin Gao, and Hui Ping Yang. "Structure Parameter Identification of Articulated Arm Coordinate Measuring Machines Based on the Least Squares Method." Applied Mechanics and Materials 303-306 (February 2013): 638–41. http://dx.doi.org/10.4028/www.scientific.net/amm.303-306.638.

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The AACMM is a kind of new coordinate measuring instrument based on non-Cartesian system. The errors of structure parameters are the main factors which affect the measuring accuracy, and the structural parameters identification can effectively improve the measuring accuracy of AACMM. The measuring model of an AACMM was established using the D-H method. Then the error model of the AACMM was established with total differential transforming method, from which the Jacobian matrix was obtained. The structure parameters’ errors which need to be identified through singular value decomposition of Jacobian matrix and the decomposition of orthogonal matrix elementary row transform can be determined. Finally, the least squares method was used in identifying the structure parameters’ errors of the AACMM, the results shows that the method proposed in this paper can identify the structure parameters efficiently.
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Cvetković, Dragana, Đorđe Vukelić, and Ksenija Doroslovački. "A New Subclass of H-Matrices with Applications." Mathematics 12, no. 15 (2024): 2322. http://dx.doi.org/10.3390/math12152322.

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The diagonal dominance property has been applied in many different ways and has proven to be very useful in various research areas. Its generalization, also known under the name H-matrix property, can be applied and produce significant benefits in economic theory, environmental sciences, epidemiology, neurology, engineering, etc. For example, it is known that the (local) stability of a (nonlinear) dynamic system is ensured if the (Jacobian) matrix belongs to the H-matrix class, and all its diagonal elements are negative. However, checking the H-matrix property itself is too expensive (from a computational point of view), so it is always worth investing effort in finding new subclasses of H-matrices, defined by relatively simple and practical conditions. Here, we will define a new subclass, which is closely related to the Euclidean vector norm, give some possible applications of this new class, and consider its relationship to some known subclasses.
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Meng, Guang Zhu, Ling Yu Sun, Ping Peng, Xian Chun Meng, Hong Mei Wang, and Jian Wei Zhang. "Jacobian Matrix of a Novel Continuum Robot for Search and Rescue." Applied Mechanics and Materials 303-306 (February 2013): 1695–701. http://dx.doi.org/10.4028/www.scientific.net/amm.303-306.1695.

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In this paper, a novel continuum robot for search and rescue is presented. A forward kinematic model is derived by product of exponentials formula, compare with conventional D-H method, this method is concise and simplicity. Finally, based on the differential kinematics using the chain rule, the overall Jacobian of the robot is established. This approach can be generally applied to various continuum robots, regardless of the specific actuation system used.
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Book chapters on the topic "Jacobian matrix H"

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Chan, Raymond H., Chen Greif, and Dianne P. O’Leary. "(With C. De Boor)." In Milestones In Matrix Computation: Selected Works Of Gene H. Golub, With Commentaries. Oxford University PressOxford, 2007. http://dx.doi.org/10.1093/oso/9780199206810.003.0030.

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Abstract This was a fun paper to work on. Gene had come to the Mathematics Research Center (MRC) in Madison, WI, in May of 1976 (giving Gene the pleasure to write in the paper’s acknowledgements of having been a “guest worker” there) to work with Paul Concus, had recently talked with Victor Barcilon about the inverse eigenvalue problem for 5-diagonal matrices, and must have given a talk about the general problem, with stress on tridiagonal matrices. I became intrigued since Gene mentioned some unsolved problems and I had recently gone through, jointly with Allan Pinkus, Gantmakher and Krein’s “Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme”, hence felt at home with Jacobi matrices and the power of Sylvester’s determinant identity (which, so Allan had taught me, is the only determinant identity one really needs, other than Cauchy–Binet).
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Conference papers on the topic "Jacobian matrix H"

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Lee, Chung-Ching, Jeng-Hong Chiu, and Hung-Hui Wu. "Kinematics of a H-Type Pure Translational Parallel Manipulator." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84992.

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From the viewpoint of kinematics, a three-dof H-type pure translational parallel manipulator is presented for the development of automatic assembly devices and a regional structure of a six-dof hybrid parallel platform. First, we describe the structural properties of manipulator and analyze its kinematic mobility. A pure translational motion is verified to exist through the well-known D-H symbolic notations and the coordinate transformation technique. What follows are the forward and inverse kinematic analysis and their analytical closed-form solutions by the matrix algebra approach. For further confirmation of the derived equations, some numerical examples are also given. Moreover, with the help of the analytical displacement kinematics, we identify the volume of workspace. At last, taking account of the overall Jacobian matrix provides the condition number and the identification of singularity of configuration is explored based on the direct and inverse kinematics Jacobian matrix.
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Young, John C., Robert J. Adams, and Stephen D. Gedney. "Efficient Jacobian Matrix Determination for $H^{2}$ Representations of Nonlinear Electrostatic Surface Integral Equations." In 2020 International Applied Computational Electromagnetics Society Symposium (ACES). IEEE, 2020. http://dx.doi.org/10.23919/aces49320.2020.9196119.

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Lee, Chung-Ching, and Po-Chih Lee. "Kinematics of a Prism-Type Translational Robot." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86714.

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From the viewpoint of kinematics, a three-dof Prism-type pure translational parallel robot is presented for the development of automatic assembly devices and a regional structure of a six-dof hybrid parallel platform. First, we describe the structural properties of the robot and analyze its kinematic mobility. A pure translational motion is verified to exist through the well-known D-H symbolic notations and the coordinate-transformation-matrix technique. What follows are the forward, inverse kinematics analysis, and their closed-form solutions by the matrix algebra approach. For further confirmation of the correctness of derived equations, some numerical examples are also given. With the help of the analytical displacement kinematics, we identify the volume of workspace. At last, taking account of the 3×3 reduced Jacobian matrix provides the condition number and the identification of singularity of configuration is explored based on the direct and inverse kinematics Jacobian matrix.
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Butuk, Nelson, and JeanPaul Pemba. "Computing CHEMKIN Sensitivities Using Complex Variables." In ASME 2001 Engineering Technology Conference on Energy. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/etce2001-17013.

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Abstract This paper discusses an accurate numerical approach of computing the Jacobian Matrix for the calculation of low dimensional manifolds for kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a highly stiff reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size, h.
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Lee, Chung-Ching, and Jeng-Hong Chou. "Kinematics of a 3-DOF Translational Parallel Mechanism With 3-PRPaR Topology." In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/detc2004-57193.

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From the standpoint of kinematics, we present a type of three-dof pure spatial translational parallel mechanism with 3-PRPaR topology as an alternative to design automation devices and a regional structure of a six-dof hybrid parallel platform. First, we describe the structural properties of mechanism and analyze its kinematic mobility. It is verified that a pure translational motion does exist through the coordinate transformation technique and the well-known D-H parametric notations. Then, we proceed with the forward and inverse kinematic analysis and derive their analytical closed-form solutions by the matrix algebra method. For the confirmation of the derived equations, some numerical examples are also taken. Furthermore, with the help of the forward kinematics, we derive the workspace in the analytical form. Finally, taking account of the overall Jacobian matrix provides the condition number and the identification of singular configuration is explored based on direct and inverse kinematics Jacobian matrix.
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Lee, Chung-Ching, and Po-Chih Lee. "Kinematics of a Pure Translational UPS/3RPaPaR Parallel Mechanism." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34930.

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From the viewpoint of kinematics, a type of 3 degrees of freedom (dofs) UPS/3RPaPaR overconstrained parallel mechanism (Pa means the hinged 4R parallelogram) with pure translational motion is presented for the development of automatic assembly devices or as a regional structure in the hybrid parallel platform. In the beginning, the formation & mobility are elucidated and the 4×4 transformation matrix & the D-H notation with specific geometric constraints verify the pure translational motion. The forward and inverse kinematic analyses are then established in the analytical closed-form through the matrix method. Besides, we take a numerical illustration for the confirmation of correctness of the derived equations. The determination of workspace is also attained by the intersection of volumes swept by each limb. In addition, the Jacobian matrix and its condition number indicated by Euclidean norm as a function of design parameters are further achieved. Finally, the singularity analysis of the configuration based on the direct and inverse kinematic J-matrix during the movement is identified in detail.
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Anno, Kotaro, George J. Moridis, and Thomas A. Blasingame. "JFTS+H: A Julia-Based Parallel Simulator for the Description of the Coupled flow, Thermal and Geochemical Processes in Hydrate-Bearing Geologic Media." In SPE Reservoir Simulation Conference. SPE, 2021. http://dx.doi.org/10.2118/203953-ms.

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Abstract The objectives of this study are to develop (a) the Julia Flow and Transport Simulator (JFTS), a serial and parallel, high performance non-isothermal, multi-phase, multi-component general simulator of flow and transport through porous/fractured media, and (b) an associated module that describes quantitatively the Equation-of-State (EOS) of the complete H2O+CH4 system by covering all combinations of phase coexistence that are possible in geologic media and including all the regions of the phase diagram that involve CH4-hydrates. The resulting simulator (hereafter referred to as the JFTS+H code) can describe all possible scenarios of hydrate occurrence, dissociation and formation/evolution and is to be used for the investigation of problems of (a) gas production from natural CH4-hydrate accumulations in geologic media, as well as for (b) the analysis of any laboratory experiments involving CH4-hydrates. As indicated by the JFTS name, this simulator is written in the Julia programming language and its parallelization is based on the Message Passing Interface (MPI) approach. The JFTS+H simulator is a fully-implicit, Jacobian-based compositional simulator that describes the accumulation, flow and transport of heat, and up to four mass components (H2O, CH4, CH4-hydrate and a water-soluble inhibitor) distributed among four possible phases (aqueous, gas, hydrate, and ice) in complex 3D geologic systems. The dissociation and formation of CH4-hydrates can be described using either an equilibrium or a kinetic model. The automatic derivate capability of Julia greatly simplifies and enhances the Jacobian computations. The MPI Interface (Blaise, 2019) is implemented in all components of the code, and the METIS library (Karypis, 2013) is used for the domain decomposition needed for the effective parallelization of the solution of the Jacobian matrix equation that is accomplished using the LIS library (Nishida, 2010) of parallel Conjugate Gradient solvers for large systems of simultaneous linear equations. The JFTS+H code can model the fluid flow, thermal and geochemical processes associated with the formation and dissociation of CH4-hydrates in geological media, either in laboratory or in natural hydrate accumulations. This code can simulate any combination of the three possible gas hydrate dissociation methods (depressurization, thermal stimulation, and inhibitor effects), and computes all associated parameters describing the system behavior. The JFTS+H results show very good agreement with solutions of standard reference problems, and of large 2D and 3D problems obtained from another well-established and widely used numerical simulator. The code exploits the speed, computational efficiency and low memory requirements of the Julia programming language. The parallel architecture of JFTS+H addresses the persistent problem of very large computational demands in serial hydrate simulations by using multiple processors to reduce the overall execution time and achieve scalable speedups. The code minimizes communications between processors and maximizes computations within the same computational node, which has important consequences (especially when coupled with the automatic derivative capabilities of Julia) on performance in the development of the Jacobian matrix. An optimal LIS solver is recommended for this type of problem after evaluating different options. This approach provides both speedup and computational efficiency results when different numbers of processors are called in the solution process. This work is believed to be the first application of Julia (a new, highly efficient language designed for demanding scientific computations) to create a simulator for flow and transport in porous media. JFTS+H is a fast, robust parallel simulator that uses the most recent scientific advances to account for all known processes in a dynamic hydrate system and works seamlessly on any computational platform (from laptop computers to workstations, to clusters and supercomputers with thousands of processors).
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Mehraban, Arash, Jed Brown, Henry Tufo, Jeremy Thompson, Rezgar Shakeri, and Richard Regueiro. "Efficient Parallel Scalable Matrix-Free 3D High-Order Finite Element Simulation of Neo-Hookean Compressible Hyperelasticity at Finite Strain." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-70768.

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Abstract The paper investigates matrix-free high-order implementation of finite element discretization with p-multigrid preconditioning for the compressible Neo-Hookean hyperelasticity problem at finite strain on unstructured 3D meshes in parallel. We consider two formulations for the matrix-free action of the Jacobian in Neo-Hookean hyperelasticity: (i) working in the reference configuration to define the second Piola-Kirchhoff tensor as a function of the Green-Lagrange strain S(E) (or equivalently, the right Cauchy-Green tensor C = I+2E), and (ii) working in the current configuration to define the Kirchhoff stress in terms of the left Cauchy-Green tensor τ(b). The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. We utilize p-multigrid preconditioning on the high-order problem with algebraic multigrid (AMG) on the assembled linear Q1 coarse grid operator. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order p, instead of the element size h. A Chebyshev polynomial smoother is used on each multigrid level. AMG is then applied to the assembled Q1 (trilinear hexahedral elements), which allows low storage that can be efficiently used to accelerate convergence to a solution. For the compressible Neo-Hookean hyperelastic constitutive model we exploit the stored energy density function to compute the stored elastic energy density of the Neo-Hookean material as it relates to the deformation gradient. Based on our formulation, we consider four different algorithms each with different storage strategies. Algorithms 1 and 3 are implemented in the reference and current configurations respectively and store ∇Xξ, det(∇ξX), and ∇Xu. Algorithm 2 in the reference configuration stores, ∇Xξ, det(∇ξX), ∇Xu, C−1, and λ log (J). Algorithm 4, in the current configuration, stores det(∇ξX), ∇xξ, τ, and μ – λ log(J). x refers to the current coordinates, X to the reference coordinates, and ξ to the natural coordinates. We perform 3D bending simulations of a tube composed of aluminum (modulus of elasticity E = 69 GPa, Poisson’s ratio v = 0.3) using unstructured meshes and polynomials of order p = 1 through p = 4 under mesh refinement. We explore accuracy-time-cost tradeoffs for the prediction of strain energy across the range of polynomial degrees and Jacobian representations. In all cases, Algorithm 4 using the current configuration formulation outperforms the other three algorithms and requires less storage. Similar simulations for large deformation compressible Neo-Hookean hyperelasticity as applied to the same aluminum material are conducted with ABAQUS, a commercial finite element software package which is a state-of-the-art engineering software package for finite element simulations involving large deformation. The best results from the proposed implementations and ABAQUS are compared in the case of p = 2 on an Intel system with @2.4 GHz and 128 GB RAM. Algorithm 4 outperforms ABAQUS for polynomial degree p = 2.
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Zhao, Yong Sheng, Kui Jing Zheng, Qin Chuan Li, Xiao Jing Tian, and Yan Ming Qin. "Kinematic Performance of 5-UPS/PRPU 5-Axis Parallel Machine Tool." In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/detc2004-57029.

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A novel 5-UPS/PRPU 5-axis PMT (Parallel Machine Tool) is studied. The stationary platform is connected with the moving platform by the same five UPS actuated limbs and a PRPU passive constraining limb. Three translational DOF (degrees of freedom) and two rotational DOF can be achieved. by using kinematic screw theory and D-H parameter method, the kinematic forward and inverse solutions of the PRPU constraining limb are deduced. The rotational constraint along the vertical axis of the moving platform acted by the PRPU limb is confirmed. Moreover, the configuration of the moving platform can be acquired online by installing sensors on the joints of the constraining limb. The kinematic inverse solution equation and jacobian matrix for this 5-axis PMT are presented. Finally the workspace and the dexterity of the PMT are analyzed. Theoretical analysis is proved by the actual operation of the prototype of the 5-UPS/PRPU 5-axis PMT in our laboratory.
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Mehraban, Arash, Jed Brown, Valeria Barra, Henry Tufo, Jeremy Thompson, and Richard Regueiro. "Efficient Residual and Matrix-Free Jacobian Evaluation for Three-Dimensional Tri-Quadratic Hexahedral Finite Elements With Nearly-Incompressible Neo-Hookean Hyperelasticity Applied to Soft Materials on Unstructured Meshes in Parallel, With PETSc and libCEED." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-24522.

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Abstract Soft materials such as rubber, elastomers, and soft biological tissues mechanically deform at large strain isochorically for all time, or during their initial transient (when a pore fluid, typically incompressible such as water, does not have time to flow out of the deforming polymer or soft tissue porous skeleton). Simulating these large isochoric deformations computationally, such as with the Finite Element Method (FEM), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation typically perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEM solid mechanics solver must run efficiently and stably. Stability is ensured by the higher order finite element formulation (with possible enhancement), but efficiency for higher order FEM remains a challenge. Thus, this paper addresses efficiency from the perspective of computer science algorithms and programming. The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. For preconditioning, a scalable p-multigrid method is presented whereby a hierarchy of levels is constructed. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order, p, instead of the element size, h. A Chebyshev polynomial smoother is used on each multigrid level. Algebraic MultiGrid (AMG) is then applied to the assembled Q1 (linear) coarse mesh on the nodes of the quadratic Q2 (quadratic) mesh. This allows low storage that can be efficiently used to accelerate the convergence to solution. For a Neo-Hookean hyperelastic problem, we examine a residual and matrix-free Jacobian formulation of a tri-quadratic hexahedral finite element with enhancement. Efficiency estimates on AVX-2 architecture based on CPU time are provided as a comparison to similar simulation (and mesh) of isochoric large deformation hyperelasticity as applied to soft materials conducted with the commercially-available FEM software program ABAQUS. The particular problem in consideration is the simulation of an assistive device in the form of finger-bending in 3D.
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