Artificial Stiffness Research Articles

Reduction of the model prediction error in vibration-based SHM applications

The accuracy of an SHM scheme is partly controlled by the accuracy of the predictive models involved within the design process of the detector that associates sensor data with a corresponding health state. On the other hand, the accuracy of the predictive model, which is quantified by the error between predictions and observations, is partly controlled by its parameter magnitudes. This work provides a unified framework that allows for reducing the model prediction error by updating the model parameters in the light of experimental observations. The problem is cast in a probabilistic setting and tackled through inferential statistics. Forward and inverse uncertainty quantification approaches are employed in a sequential manner. The framework is able to yield point estimates in accordance to a Maximum Likelihood Estimator (MLE) and is also able to construct credible intervals through computational Bayesian updating. The former is treated by genetic algorithms whereas the later through Markov Chain Monte Carlo computational statistics. The framework is showcased on a vibration problem of the simplified benchmark case named as “Jim Beam”. The first ten modal parameters are considered as the discriminant features of interest . A surrogate of the finite element model of the considered specimen serves as the predictive model and the eigenfrequencies are extracted from a linear modal analysis. The parameters of inferential interest are the Young’s modulus and an artificial stiffness related parameter that is introduced so as to model the flexural rigidity of the specimen’s bolted connections. Experimental Modal Analysis and Operational Modal Analysis are the two approaches that were employed for experimentally identifying the modes and their corresponding frequencies. Results are provided from both the MLE and the Bayesian updating process and the effectiveness of each approach is discussed.

Estimation of Knee Assistive Moment in a Gait Cycle Using Knee Angle and Knee Angular Velocity through Machine Learning and Artificial Stiffness Control Strategy (MLASCS)

Nowadays, many people around the world cannot walk perfectly because of their knee problems. A knee-assistive device is one option to support walking for those with low or not enough knee muscle forces. Many research studies have created knee devices with control systems implementing different techniques and sensors. This study proposes an alternative version of the knee device control system without using too many actuators and sensors. It applies the machine learning and artificial stiffness control strategy (MLASCS) that uses one actuator combined with an encoder for estimating the amount of assistive support in a walking gait from the recorded gait data. The study recorded several gait data and analyzed knee moments, and then trained a k-nearest neighbor model using the knee angle and the angular velocity to classify a state in a gait cycle. This control strategy also implements instantaneous artificial stiffness (IAS), a control system that requires only knee angle in each state to determine the amount of supporting moment. After validating the model via simulation, the accuracy of the machine learning model is around 99.9% with the speed of 165 observers/s, and the walking effort is reduced by up to 60% in a single gait cycle.

New transverse extension modes for buckling analysis of thin-walled members by using the constrained finite strip method

In this paper a new way to define transverse extension modes is introduced in the context of modal analysis of thin-walled members. The new transverse extension modes are inspired by the strain distributions of global and distortional modes. The new modes are employed here in the constrained finite strip method. The new modes require some change in the interpolation functions of the semi-analytical finite strip method, and this change is briefly discussed. The practical advantage of the newly proposed transverse extension modes is that the artificial stiffness increase, which is characteristic to pure global and/or distortional deformations in constrained analyses, can be compensated by the consideration of the relevant new transverse extension mode(s), as illustrated by numerous proof-of-concept examples in the paper.

Design and evaluation of an active secondary mirror positioning system for a small telescope

This publication presents an active compensation system for the position of the secondary mirror of a small Ritchey–Chrétien telescope system. The goal is to maintain the optical imaging quality under varying gravitational and thermal influences, by compensation for the relative position deviations between the primary and secondary mirrors. An extensive analysis concerning the feasibility of such a system for a commercial off-the-shelf small telescope is performed and used as a basis for the design of the precision measurement and positioning system. The developed prototype uses dimensional metrology to capture relative position errors of the secondary mirror. A newly designed actuator with three degrees of freedom for the secondary mirror allows us to compensate for these deviations in a closed-loop control manner and ensures optimal positions of the two mirrors at all times. The support structure design requirements are reduced, allowing the utilization of more lightweight structures, as the artificial stiffness of the compensation system takes care of keeping the telescope mirrors in place. Furthermore, the measurement principle requires no light from the telescope, thus providing 100% of the collected light for the observation. The developed actuation and measurement principles are designed for simple scalability to larger representatives of small telescopes. The implemented setup is evaluated in various poses and temperature influences, successfully demonstrating that the calculated Strehl ratio is kept well above the diffraction limit of 80% for the used telescope system.

Provably non-stiff implementation of weak coupling conditions for hyperbolic problems

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant–Friedrichs–Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals.

Research on solid shell element based on penalty method

Since the angle degrees of freedom do not have the vector property, it is critical to handle this for nonlinear geometrical behavior. To simulate the nonlinear geometrical behavior of shell structure efficiently, we developed a solid‐shell element type without drilling degrees of freedom. In this paper, the element satisfies the assumption of the straight normal of the shell and we established penalty functions in the thickness direction of the shell. The formulations of equivalent node force and tangent stiffness obtained by 3D isoparametric element theory can be expressed by matrix forms clearly. The constitutive relationship in the global system is derived from the local constitutive matrix according to the tensor coordinate transform. Finally, taking a series of classic flat plate problems and curved shell problems as examples, the solid shell element using the penalty method is compared with the solid shell element using the artificial stiffness method and other shell elements. The results show that the solid element proposed in this paper can effectively eliminate shear self-locking after using the reduced integral scheme and obtain satisfactory accuracy for thin shells and medium-thickness•A type of solid shell element without rotational degrees of freedom is introduced using the penalty method.•A suitable penalty factor of this element is determined through numerical experiments.•This element is more efficient than the traditional shell elements and can be programmed with less time.

Stepped and swept control-based continuation using adaptive filtering

This paper introduces a new online method for performing control-based continuation (CBC), speeding up the model-less identification of stable and unstable periodic orbits of nonlinear mechanical systems. The main building block of the algorithm is adaptive filtering which can ensure the non-invasiveness of the controller without the need for offline corrective iterations. Two different strategies, termed stepped and swept CBC, are then developed for performing the continuation steps. A beam featuring different artificial stiffness and damping nonlinearities is considered for the experimental demonstration of the proposed developments. The performance of the CBC strategies are compared in terms of running time and identification accuracy.

Real-Time Simulation for Control of Soft Robots With Self-Collisions Using Model Order Reduction for Contact Forces

In rigid robotics, self-collision are usually avoided since it leads to a failure in the robot control and can also cause damage. In soft robotics, the situation is very different, and self-collisions may even be a desirable property, for example to gain artificial stiffness or to provide a natural limitation to the workspace. However, the modeling and simulation of self-collision is very costly as it requires first a collision detection algorithm to detect where collisions occur, and most importantly, it requires solving a constrained problem to avoid interpenetrations. When the number of contact points is large, this computation slows down the simulation dramatically. In this letter, we apply a numerical method to alleviate the contact response computation by reducing the contact space in a low-dimensional positive space obtained from experiments. We show good accuracy while speeding up dramatically the simulation. We apply the method in simulation on a cable-actuated finger and on a continuum manipulator performing exploration. We also show that the reduced contact method proposed can be used for inverse modeling. The method can therefore be used for control or design.

New Transverse Extension Modes for the Constrained Finite Strip Analysis of Thin-walled Members

In this paper a new way to define transverse extension modes is introduced in the context of modal analysis of thin-walled members. The new primary transverse extension modes are inspired by the strain distributions of global and distortional modes. The new modes are employed here in the constrained finite strip method. The new modes require some change in the interpolation functions of the semianalytical finite strip method, which change is briefly discussed. The practical advantage of the proposed new transverse extension modes is that the artificial stiffness increase, which is characteristics to constrained analyses, can be compensated by the consideration of the relevant new transverse extension mode(s), as illustrated by numerous proof-of-concept examples in the paper.

A DG-based interface element method for modeling hydraulic fracturing in porous media

Fracture propagation coupled with fluid flow in porous media has very important applications in fracking in oil and gas reservoirs, corrosion in ceramics, and degradation of human bones. Modeling fracture failure in porous media governed by the coupled solid and fluid field equations is challenging and time-consuming. This paper extends discontinuous Galerkin (DG) finite element methods to model crack openings in porous media through exploiting an easy construction of interfaces for potential crack paths by locally breaking continuous elements. Consequently, a finite element mesh for fracture apertures is completely and gracefully constructed using DG interface objects as well as the definitions of jumps of displacements across the DG interfaces defined for bulk matrices. Furthermore, rigid cohesive laws often adopted for crack openings can be naturally implemented in DG formulations without introducing artificial stiffness, which may eventually improve the performance for implicit formulations in handling crack openings in porous media. In this work, we perform a consistent, fully implicit, and fully coupled hybrid DG/continuous finite element formulation for three field equations including the solid, bulk fluid, and fluid in fracture apertures resulting from crack openings. We verify our DG formulation and implementation using the Khristianovich-Geertsma-DeKlerk analytical solutions. Finally, we further demonstrate a good performance of the proposed DG method by modeling fracking in an oil reservoir containing two nature fractures.

A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method

This paper presents a novel computational approach for SIMP-based Topology Optimisation (TO) of hyperelastic materials at large strains. During the TO process for structures subjected to very large deformations, and especially in the presence of intermediate density regions, the standard Newton-solver (or its arc length variant) have been reported not to converge (refer to References Wang et al. (2014), Lahuerta et al. (2013) and Liu et al. (2017)). In this paper, the new TO stabilisation technique proposed in Ortigosa et al. (2019) in the context of level-set TO, initially devised to alleviate numerical instabilities inherent to level-set TO, is extended for the TO by means of the SIMP method. The success of the methodology rests on the combination of two distinct key ingredients. First, the nonlinear equilibrium equations of motion for intermediate TO design stages are solved in a non-exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is locally stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. This solution strategy is shown to be extremely robust in the context of density-based TO, where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents, the latter classically defined by means of a fictitious strain energy. The robustness and applicability of this TO methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the final designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton–Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10−12.

Strain localization in reduced-order asymptotic homogenization

A reduced-order asymptotic homogenization-based multiscale technique that can capture damage and inelastic effects in composite materials is proposed. This technique is based on a two-scale homogenization procedure where eigenstrain representation accounts for the inelastic response and the computational efforts are alleviated by a reduction-of-order technique. Macroscale stress is derived by calculating the influence tensors from analysis of a representative volume element. At microscale, the damage in the material is modeled using a framework based on continuum damage mechanics. To solve the problem of strain localization, a method of alteration of the stress–strain relation of microconstituents based on the dissipated fracture energy in a crack band is implemented. The issue of spurious postfailure artificial stiffness at macroscale is discussed and the effect of increasing the order to alleviate this problem is checked. Verification studies demonstrated that the proposed formulation predicts the macroscale response and also captures the damage- and plasticity-induced inelastic strains.

Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

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 new stabilisation approach for level-set based topology optimisation of hyperelastic materials

This paper introduces a novel computational approach for level-set based topology optimisation of hyperelastic materials at large strains. This, to date, is considered an unresolved open problem in topology optimisation due to its extremely challenging nature. Two computational strategies have been proposed to address this problem. The first strategy resorts to an arc-length in the pre-buckling region of intermediate topology optimisation (TO) iterations where numerical difficulties arise (associated with nucleation, disconnected elements, etc.), and is then continued by a novel regularisation technique in the post-buckling region. In the second strategy, the regularisation technique is used for the entire loading process at each TO iteration. The success of both rests on the combination of three distinct key ingredients. First, the nonlinear equilibrium equations of motion are solved in a consistent incrementally linearised fashion by splitting the design load into a number of load increments. Second, the resulting linearised tangent elasticity tensor is stabilised (regularised) in order to prevent its loss of positive definiteness and, thus, avoid the loss of convexity of the discrete tangent operator. Third, and with the purpose of avoiding excessive numerical stabilisation, a scalar degradation function is applied on the regularised linearised elasticity tensor, based on a novel regularisation indicator field. The robustness and applicability of this new methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples, ranging from benchmark two-dimensional (plane stress) examples to larger scale three-dimensional applications. Crucially, the performance of all the designs has been tested at a post-processing stage without adding any source of artificial stiffness. Specifically, an arc-length Newton-Raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 10− 12.

Differential-Geometric-Control Formulation of Flapping Flight Multi-body Dynamics

Flapping flight dynamics is quite an intricate problem that is typically represented by a multi-body, multi-scale, nonlinear, time-varying dynamical system. The unduly simple modeling and analysis of such dynamics in the literature has long obstructed the discovery of some of the fascinating mechanisms that these flapping-wing creatures possess. Neglecting the wing inertial effects and directly averaging the dynamics over the flapping cycle are two major simplifying assumptions that have been extensively used in the literature of flapping flight balance and stability analysis. By relaxing these assumptions and formulating the multi-body dynamics of flapping-wing micro-air-vehicles in a differential-geometric-control framework, we reveal a vibrational stabilization mechanism that greatly contributes to the body pitch stabilization. The discovered vibrational stabilization mechanism is induced by the interaction between the fast oscillatory aerodynamic loads on the wings and the relatively slow body motion. This stabilization mechanism provides an artificial stiffness (i.e., spring action) to the body rotation around its pitch axis. Such a spring action is similar to that of Kapitsa pendulum where the unstable inverted pendulum is stabilized through applying fast-enough periodic forcing. Such a phenomenon cannot be captured using the overly simplified modeling and analysis of flapping flight dynamics.

Static behavior of a laterally loaded guardrail post in sloping ground by LS-DYNA

This study aims to present accurate soil modeling and validation of a single roadside guardrail post as well as a single concrete pile installed near cut slopes or compacted sloping embankment. The conventional Winkler\'s elastic spring model and p-y curve approach for horizontal ground cannot directly be applied to sloping ground where ultimate soil resistance is significantly dependent on ground inclination. In this study, both grid-based 3-D FE model and particle-based SPH (smoothed particle hydrodynamics) model available in LS-DYNA have been adopted to predict the static behavior of a laterally loaded guardrail post. The SPH model has potential to eliminate any artificial soil stiffness due to the deterioration of the node-connected Lagrangian soil mesh. For this purpose, this study comprises two parts. Firstly, only 3-D FE modeling has been tested to show the numerical validity for a single concrete pile in sloping ground using Mohr-Coulomb material. However, this material option cannot be implemented for SPH elements. Nevertheless, Mohr-Coulomb model has been used since this material model requires six input soil data that can be obtained from the comparative papers in literatures. Secondly, this work is extended to compute the lateral resistance of a guardrail post located near the slope using the hybrid approach that combines Lagrange FE elements and SPH elements by the suitable node-merging option provided by LS-DYNA. For this analysis, the FHWA soil material developed for application to road-base soils has been used and also allows the application of SPH element.

On the Scale-Transition in Multiscale Modeling of Ductile Damage

The standard numerical methods cannot adequately capture damage response of ductile heterogeneous materials, since they are unable to encompass the governing microstructural mechanisms. This issue is resolved by the multiscale approach, where the coarse scale problem incorporates behavior at the lower scales. In this paper, a multiscale scheme is employed for scaling of ductile damage occurring at the microstructural level to the macrolevel. A nonlocal implicit damage model is applied on the RVE, while a scale transition is performed by means of the first-order homogenization scheme. The influence of the boundary conditions used at the microscale on the macroscale response, as well as physical consistency of the RVE mechanical behavior are tested. It has been shown that displacement boundary conditions inhibit development of the softening band inside of the RVE and induce artificial material stiffness. The derived algorithms are embedded into the finite element software ABAQUS.

Material nonlinear topology optimization using the ground structure method with a discrete filtering scheme

Topology optimization of truss lattices, using the ground structure method, is a practical engineering tool that allows for improved structural designs. However, in general, the final topology consists of a large number of undesirable thin bars that may add artificial stiffness and degenerate the condition of the system of equations, sometimes even leading to an invalid structural system. Moreover, most work in this field has been restricted to linear material behavior, yet real materials generally display nonlinear behavior. To address these issues, we present an efficient filtering scheme, with reduced-order modeling, and demonstrate its application to two- and three-dimensional topology optimization of truss networks considering multiple load cases and nonlinear constitutive behavior. The proposed scheme accounts for proper load levels during the optimization process, yielding the displacement field without artificial stiffness by simply using the truss members that actually exist in the structure (spurious members are removed), and improving convergence performance. The nonlinear solution scheme is based on a Newton-Raphson approach with line search, which is essential for convergence. In addition, the use of reduced-order information significantly reduces the size of the structural and optimization problems within a few iterations, leading to drastically improved computational performance. For instance, the application of our method to a problem with approximately 1 million design variables shows that the proposed filter algorithm, while offering almost the same optimized structure, is more than 40 times faster than the standard ground structure method.

Minimizing Artificial Stiffness in Linear Tetrahedral Element Using Virtual Mesh Refinement

AbstractThis work presents a new method to minimize artificial stiffness in linear tetrahedral element using virtual mesh refinement (VRM) method. The basic idea behind this work is to give additional degree of freedom by using internal mesh over the linear tetrahedral element. This local internal mesh and its corresponding equilibrium condition under particular boundary condition are invisible to users or virtual. Using specialized displacement test vectors, strain energy is obtained and used to calculate reduction factor for artificial stiffness. Numerical experiments are performed at the end to briefly qualitatively show performance of our proposed method.

Use of Measured Vibration ofIn-SituSleeper for Detecting Underlying Railway Ballast Damage

The identification of railway ballast damage under a concrete sleeper is investigated by following the Bayesian approach. The use of a discrete modeling method to capture the distribution of ballast stiffness under the sleeper introduces artificial stiffness discontinuities between different ballast regions. This increases the effects of modeling errors and reduces the accuracy of the ballast damage detection results. In this paper, a continuous modeling method was developed to overcome this difficulty. The uncertainties induced by modeling error and measurement noise are the major difficulties of vibration-based damage detection methods. In the proposed methodology, Bayesian probabilistic approach is adopted to explicitly address the uncertainties associated with the identified model parameters. In the model updating process, the stiffness of the ballast foundation is assumed to be continuous along the sleeper by using a polynomial of order N. One of the contributions of this paper is to calculate the order N conditional on a given set of measurement utilizing the Bayesian model class selection method. The proposed ballast damage detection methodology was verified with vibration data obtained from a segment of full-scale ballasted track under laboratory conditions, and the experimental verification results are very encouraging showing that it is possible to use the Bayesian approach along with the newly developed continuous modeling method for the purpose of ballast damage detection.