Internal DoFs Research Articles

Rate-dependent augmented finite element method for fast arbitrary crack growth

Experimental observations indicate that fast crack growth often exhibits a rate dependency, which is caused by the viscosity of the material and the dissipation of energy resulting from the branching of microcracks. To address this phenomenon in simulation modeling, this study introduces a rate-dependent augmented finite element method designed for analyzing fast arbitrary crack growth. The method utilizes the maximum principal stress criterion to simulate arbitrary crack growth and predict the path of crack growth. The elements where crack growth is expected are divided into subelements, and they are re-assembled by incorporating rate-dependent cohesive elements between them to construct augmented finite elements. The cohesive elements are based on a rate-dependent trapezoid traction–separation relationship, where the critical traction and fracture toughness are modeled as functions of the separation rate and crack growth speed. The augmented finite elements only contain external nodal degrees of freedom (DoFs), while internal nodal DoFs are condensed. They can be utilized as conventional elements. A numerical algorithm based on the proposed method was developed in this article, and it was implemented in ABAQUS via user element subroutine. The effectiveness of this method and algorithm is validated through an example of dynamic crack growth in a double cantilever beam. The results demonstrate agreement with experimental and computational findings reported in the literature. Additionally, the method was applied to other structures, yielding reasonable outcomes. Analysis of these examples confirms the method’s utility in predicting fast arbitrary crack growth under dynamic load conditions.

A critical evaluation of SEMM-based joint identification procedure to reduce the error propagation effects

Mechanical joints play a critical role in many engineering structures, but identifying their characteristics can be challenging, particularly when the joint interface (coupling DoFs) cannot be directly measured. In these cases, an existing iterative procedure based on the use of the System Equivalent Model Mixing (SEMM) expansion technique and substructure decoupling can be used to identify the joint properties by taking measurement on other accessible DoFs (internal DoFs). Despite the potential of this procedure, it is prone to large error propagation. In addition, in the SEMM expansion a weighted pseudo-inverse operation is needed to ensure the convergence of the iterative procedure when both coupling and internal DoFs (extended interface) are involved in the decoupling.This paper focuses on the detection of the error sources in the process and on the definition of some strategies to limit error propagation. The use of internal DoFs only (pseudo-interface) in the decoupling is proposed. This avoids the use of coupling DoFs affected by the expansion error. Furthermore, two strategies are proposed to improve the conditioning of the procedure when using the extended interface. Both strategies are based on the Truncated Singular Values Decomposition (TSVD). It is shown that the weights clearly indicate the number of singular values to be retained in the matrices to be inverted.The proposed improvements are validated on a laboratory benchmark. Measurements on the benchmark are performed to validate the strategies with experimental data. In addition, the Monte Carlo method is applied using noise-polluted numerical data to evaluate the potential of the proposed strategies to mitigate the error propagation in the SEMM-based joint identification procedure.

A response reconstruction method based on empirical mode decomposition and modal synthesis method

With the increasing the complexity of civil structures and diversity of the structural components, reconstructing the responses of complex civil engineering structures based on the conventional time-domain reconstruction method is very challenging. Apart from the massive computer storage needed in the calculation process, the enormous calculation times may also hinder the response reconstruction of the civil structures. This paper proposes a novel strategy of the time-domain reconstruction method by combining the modal synthesis method and empirical mode decomposition with intermittency criteria in the time domain. The response reconstruction method is conducted at two levels. The first level is performed on the reduced super-element from the interface DOFs to the internal DOFs of the critical substructure, and then at the element level to extrapolate dynamic responses at critical locations. In each level’s reconstruction, the number of calculation parameters decreases significantly. The response of the whole structure can be quickly obtained at one time, not only decreasing the computation time but also ensuring reconstruction accuracy. The numerical studies and experimental validation are both conducted on a simply supported beam and a complex cooling tower for verification, and the effects from some external parameters, including measurement noise, the order of substructural normal modes, the number of substructures and the number of sensors, are considered. The results illustrate that the proposed reconstruction method based on the modal synthesis method improves the calculation effectiveness considerably and ensures accuracy.

A novel one-degree-of-freedom translational partly compliant mechanism with variable motion direction

This study presents the design and analysis of a one-degree-of-freedom (DOF) translational partly compliant mechanism (TPCM) with variable motion direction. The variable motion direction is achieved by using rigid-body internal DOFs to change the poses of flexure beams. The use of rigid-body internal DOFs overcomes the small deflection limit of compliant joints and enables continuous changes in the motion direction. First, a simplified model is developed through mobility analysis and shows that the motion direction of the proposed TPCM can be changed continuously in a spatial manner. Then, the compliance ratio is obtained by decomposing the compliance matrix. It shows that the compliance ratio drops quickly near the singularity and that the motion direction should be limited. Finite element analysis and experiments are conducted to evaluate the design.

Replacement of unobservable coupling DoFs in substructure decoupling

In this paper, suitable criteria are sought for an optimal replacement of unobservable coupling DoFs when performing substructure decoupling, that is the identification of a dynamic model of a substructure embedded in a known structure. The need arises since coupling DoFs are often difficult to observe, either because they cannot be easily accessed or because they include rotational DoFs. The substitution must be carried out both to preserve the information that would be lost when some coupling DoFs are not taken into account, and to avoid or minimize ill-conditioning problems.As shown in previous papers, coupling DoFs can be effectively replaced by internal DoFs for the sake of substructure decoupling. Furthermore, criteria for an appropriate selection of the internal DoFs used to replace unobservable coupling DoFs are sketched, which involve either the Frequency Response Function (FRF) or the transmissibility between internal and coupling DoFs.Here, previously introduced FRF and transmissibility criteria are combined with the condition number of the interface flexibility matrix to develop a procedure to optimally replace some coupling DoFs with a subset of internal DoFs. The procedure is tested using both simulated and experimental data of a tree structure (known structure), made by a cantilever beam with two offset short arms, coupled to another beam (structure to be identified).

A two-dimensional augmented finite element for dynamic crack initiation and propagation

In this study, an implicit formulation of a 2D finite element based on a recently developed augmented finite element method is proposed for stable and efficient simulation of dynamic fracture in elastic solids. The 2D A-FE ensures smooth transition from a continuous state to a discontinuous state with an arbitrary intra-element cohesive crack, without the need of additional degree of freedoms (DoFs). Internal nodal DoFs are introduced for sub-domain integration and cohesive stress integration and they are then condensed at elemental level by a consistency-check based algorithm. The numerical performance of the proposed A-FE has been assessed through simulations of several benchmark dynamic fracture problems and in all cases the numerical results are in good agreement with the respective experimental results and other simulation results in literature. It has further been demonstrated that, (i) the dynamic A-FE is rather insensitive to mesh sizes and mesh structures; (ii) with similar solution accuracy it allows for the use of time steps 1–2 orders of magnitude larger than those used in other similar studies; and (iii) the implicit nature of the proposed A-FE allows for the use of a Courant number as large as 3.0–3.5 while maintaining solution stability.

Unique Flexibility Patterns of PDB Entries

Solid bodies obtain characteristic mass and charge distributions, principle rotational axes as well as internal symmetry axes. Internal symmetry axes are uniquely identified by Normal Mode Analysis (NMA) and are widely used in the fields of engineering and seismology to model the redistribution of modest energy influxes into non-rigid bodies. NMA of protein systems rapidly and reproduceably identify their internal flexibility characteristics. Protein systems typically consist of thousands of atoms and three times as many Cartesian degrees of freedom (dofs). These systems can equally well be characterized by their internal dofs: bond lengths, bond angles and bond rotations. Small energy perturbations like thermal baths primarily activate bond-rotations or dihedrals. Use of dihedral dofs ensures that bond lengths and bond angles remain stereochemically sound. We have shown that use of these dofs combined with distance- and atom-dependent Hookian potentials correctly reproduce theoretical dispersion spectra. Use of linearized potentials permits NMA of PDB coordinates: No initial, structure-distorting energy minimization is required. This feature permits analyses of PDB structures that differ only slightly such as (i) two isoforms of the same X-ray electron density, (ii) the same structure solved in two different crystal forms, or (iii) two evolutionary related proteins that share high structural, not sequence, homology. We examine the modal signatures of several cases and report meaningful differences. In one, glycoside hydrolase family 10 xylanases, the A and B isoforms of the same crystal place three out of 302 residues in alternate conformations. Their modal analyses reveal significant differences in their slowest modes, consistent with predictions that GH-10 members consist of two subgroups with differing mobility. In another, the modal analysis of two closely related tertiary structures with low sequence homology, lactadherin and Factor VIII glycoprotein, reveal significant differences consistent with H/D exchange data.

Substructure decoupling without using rotational DoFs: Fact or fiction?

In the framework of experimental dynamic substructuring, substructure decoupling consists in the identification of the dynamic behaviour of a structural subsystem, starting from the dynamic behaviour of both the assembled system and the residual subsystem (the known portion of the assembled system). On the contrary, substructure coupling identifies an assembled system starting from the component subsystems. The degrees of freedom (DoFs) of the assembled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs.In substructure coupling, whenever coupling DoFs include rotational DoFs, the related rotational FRFs must be obtained experimentally. Does this requirement holds for substructure decoupling too, as it is commonly believed?Decoupling can be ideally accomplished by adding the negative of the residual subsystem to the assembled system (direct decoupling) and by enforcing compatibility and equilibrium at enough interface DoFs. Ideally, every DoF of the residual subsystem belongs to the interface between the assembled system and the residual subsystem. Hopefully, not all the coupling DoFs are necessary to enforce compatibility and equilibrium. This may allow us to skip coupling DoFs and specifically rotational DoFs.The goal of the paper is indeed to establish if rotational FRFs at coupling DoFs can be neglected in substructure decoupling. To this aim, after highlighting the possibility of avoiding the use of coupling DoFs from a theoretical standpoint, a test bed coupled through flexural and torsional DoFs is considered. Experimental results are presented and discussed.

Reduced Locomotion Dynamics With Passive Internal DoFs: Application to Nonholonomic and Soft Robotics

This paper proposes a general modeling approach for locomotion dynamics of mobile multibody systems containing passive internal degrees of freedom concentrated into (ideal or not) joints and/or distributed along deformable bodies of the system. The approach embraces the case of nonholonomic mobile multibody systems with passive wheels, the pendular climbers, and the locomotion systems bioinspired by animals that exploit the advantages of soft appendages such as fish swimming with their caudal fin or moths that use the softness of their flapping wings to improve flight performance. The paper proposes a general structured modeling approach of MMS with tree-like structures along with efficient computational algorithms of the resulting equations. The approach is illustrated through nontrivial examples such as the 3-D bicycle and a compliant version of the snake-board.

An efficient augmented finite element method for arbitrary cracking and crack interaction in solids

SUMMARYThis paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary, multiple intra‐elemental discontinuities in 2D solids. The augmented finite element method (A‐FEM) employs four internal nodes to account for the crack displacements due to an intra‐elemental weak or strong discontinuity, and it permits repeated elemental augmentation to include multiple interactive cracks. It thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity, and to multiple interactive cohesive cracks, all within a single bilinear element that employs standard external nodal DoFs only. A novel elemental condensation procedure has been developed to solve the internal nodal DoFs as functions of the external nodal DoFs for any irreversible, piece‐wise linear cohesive laws. It leads to a fully condensed elemental equilibrium equation with mathematical exactness, eliminating the need for nonlinear equilibrium iterations at elemental level. The new A‐FEM's high‐fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous, and heterogeneous solids have been demonstrated through several challenging numerical examples. It is shown that the proposed A‐FEM, empowered by the new elemental condensation procedure, is numerically very efficient, accurate, and robust. Copyright © 2014 John Wiley & Sons, Ltd.

Inverse dynamic substructuring using the direct hybrid assembly in the frequency domain

The paper deals with the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of both the coupled system and the remaining part of the structural system (residual subsystem). This topic is also known as decoupling problem, subsystem subtraction or inverse dynamic substructuring. Whenever it is necessary to combine numerical models (e.g. FEM) and test models (e.g. FRFs), one speaks of experimental dynamic substructuring. Substructure decoupling techniques can be classified as inverse coupling or direct decoupling techniques. In inverse coupling, the equations describing the coupling problem are rearranged to isolate the unknown substructure instead of the coupled structure. On the contrary, direct decoupling consists in adding to the coupled system a fictitious subsystem that is the negative of the residual subsystem. Starting from a reduced version of the 3-field formulation (dynamic equilibrium using FRFs, compatibility and equilibrium of interface forces), a direct hybrid assembly is developed by requiring that both compatibility and equilibrium conditions are satisfied exactly, either at coupling DoFs only, or at additional internal DoFs of the residual subsystem. Equilibrium and compatibility DoFs might not be the same: this generates the so-called non-collocated approach. The technique is applied using experimental data from an assembled system made by a plate and a rigid mass.

Erratum to “Reduced Locomotion Dynamics With Passive Internal DoFs: Application to Nonholonomic and Soft Robotics” [Jun 14 578-592

In paper [“Reduced locomotion dynamics with passive internal DoFs: Application to nonholonomic and soft robotics,” IEEE Trans. Robot., vol. 30, no. 3, pp. 578–592, Jun. 2014; Boyer, F. and Belkhiri, A.], we proposed a general modeling framework to address locomotion systems containing internal passive degrees of freedom. Unfortunately, we committed an error in illustrating the approach on the 3-D model of the bicycle. In this erratum, we correct this error which concerns the kinematic model of the bicycle.

Equations for energy characteristics of oscillatory systems with internal (hidden) degrees of freedom and application to acoustic metamaterials

General equations are derived for calculating the kinetic and potential energies and other energy characteristics of linear oscillatory NDOF-systems a portion of DOFs of which are internal or inaccessible for measurement and excluded from consideration. The energy characteristics are expressed through parameters pertaining only to the input or accessible DOFs. The equations are based on the certain novel properties of the so-called Shur matrix complement. The theory is applied to calculating the energy characteristics of acoustic metamaterials for which this is still an unsolved problem, especially for those with negative effective density and stiffness. A metamaterial is thought as a medium or periodic structure in which the role of effective inertia and elastic elements is played by sufficiently complex oscillatory systems with internal DOFs. Applying the derived equations to a cell of periodicity of a metamaterial one can obtain the exact values of the needed energy characteristics. The theory is verified in computer simulation and laboratory experiment.

The role of interface DoFs in decoupling of substructures based on the dual domain decomposition

The paper considers the decoupling problem, i.e. the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of the coupled system, and from information about the remaining part of the structural system (residual subsystem). Typically, the FRF matrix of the coupled system is assumed to be known at the coupling DoFs (standard interface). To circumvent ill-conditioning around particular frequencies, some authors suggest the use of FRFs at some internal DoFs of the residual subsystem. In this paper, the decoupling problem is revisited in the general framework of frequency based substructuring. Specifically, the dual domain decomposition is used by adding a fictitious subsystem, which is the negative of the residual subsystem, to the coupled system. In this framework, the use of internal DoFs of the residual subsystem, in addition to coupling DoFs, appears quite natural (extended interface). The effects of using an extended interface are widely discussed: the main drawback is that the problem becomes singular at any frequency. However, this singularity is easily removed by using standard smart inversion techniques. The approach is tested on a discrete system describing a two-speed transmission, using simulated data polluted by noise. Results are compared with those obtained from existing approaches.

Thermal conduction in molecular materials using coarse grain dynamics: Role of mass diffusion and quantum corrections for molecular dynamics simulations

We use a mesodynamical method, denoted dynamics with implicit degrees of freedom (DID), to characterize thermal transport in a model molecular crystal below and above its melting temperature. DID represents groups of atoms (molecules in this case) using mesoparticles and the thermal role of the intramolecular degrees of freedom (DoFs) are described implicitly using their specific heat. We focus on the role of these intramolecular DoFs on thermal transport. We find that thermal conductivity is independent of intramolecular specific heat for solid samples and a linear relationship between the two quantities in liquid samples with the coefficient of proportionality being the mass diffusivity of the mesoparticles. As the temperature of the liquids is increased, thermal conductivity exhibits an increased sensitivity with respect to the specific heat of the internal DoFs due to the enhanced molecular mobility. Based on these results, we propose a simple method to incorporate quantum corrections to thermal conductivity obtained from nonequilibrium molecular dynamics simulations of molecular liquids. Our results also provide insight into the development of thermally accurate coarse grain models of soft materials.

Coarse grain modeling of spall failure in molecular crystals: role of intra-molecular degrees of freedom

We use a recently developed thermodynamically accurate mesodynamical method (Strachan and Holian 2005 Phys. Rev. Lett. 94 014301) where groups of atoms are represented by mesoparticles to characterize the shock compression and dynamical failure (spall) of a model molecular crystal. We characterize how the temperature rise caused by the shockwave depends on the specific heat of the degrees of freedom (DoFs) internal to the mesoparticles (Cint) and the strength of the coupling between the internal DoFs and the mesoparticles. We find that the temperature of the shocked material decreases with increasing Cint and decreasing coupling and quantify these effects. Our simulations also show that the threshold for plastic deformation (the Hugoniot elastic limit) depends on the properties of the internal DoFs while the threshold for failure is very insensitive to them. These results have implications on the results of all-atom MD simulations, whose classical nature leads to a significant overestimation of the specific heat of molecular materials.

Pose Controlled Physically Based Motion

Abstract In this paper we describe a new method for generating and controlling physically‐based motion of complex articulated characters. Our goal is to create motion from scratch, where the animator provides a small amount of input and gets in return a highly detailed and physically plausible motion. Our method relieves the animator from the burden of enforcing physical plausibility, but at the same time provides full control over the internal DOFs of the articulated character via a familiar interface. Control over the global DOFs is also provided by supporting kinematic constraints. Unconstrained portions of the motion are generated in real time, since the character is driven by joint torques generated by simple feedback controllers. Although kinematic constraints are satisfied using an iterative search (shooting), this process is typically inexpensive, since it only adjusts a few DOFs at a few time instances. The low expense of the optimization, combined with the ability to generate unconstrained motions in real time yields an efficient and practical tool, which is particularly attractive for high inertia motions with a relatively small number of kinematic constraints.