Linear Elastic Structures Research Articles

Topology optimization of structures undergoing brittle fracture

In the framework of the level-set method we propose a topology optimization algorithm for linear elastic structures which can exhibit fractures. In the spirit of Griffith theory, brittle fracture is modeled by the Francfort-Marigo energy model, with its Ambrosio-Tortorelli regularization, which can also be viewed as a gradient damage model. This quasi-static and irreversible gradient damage model is approximated using penalization to make it amenable to shape-differentiation. The shape derivative is determined using the adjoint method. The shape optimization is implemented numerically using a level-set method with body-fitted remeshing, which captures shapes exactly while allowing for topology changes. The efficiency of the proposed method is demonstrated numerically on 2D and 3D test cases. The method is shown to be efficient in conceiving crack-free structures.

A note on the use of Dynamic Mode Decomposition in mechanics

We advocate for the use of Dynamic Mode Decomposition (DMD) in mechanics. Snapshots of the temperature or displacement fields are obtained; and the DMD algorithm, which, remarkably, requires a few lines of code, is applied to matrices obtained by time shifting the snapshots. Only the higher-order modes are retained to obtain a lower-rank matrix and used for analysis. We show that DMD can be used to: predict temperature fields during transient heat conduction, estimate natural frequency of a linear elastic structure; identify the onset of plasticity in an elastic–plastic plate with a hole; and obtain a reduced-order model that describes the large deformation of a non-linear elastic structure.

Mitigation of Earthquake Responses Using SMA Supplemented Base-Isolation Devices for Benchmark Building

The efficiency of traditional isolation bearings is doubted for near-field earthquakes because these bearings undergo large displacement. A comparative study of different base isolation systems of base-isolated benchmark building is carried out in the present study. The study is based on assumption that buildings are bi-directionally acted upon by near-field earthquakes for assessing their relative performance in seismic control of the benchmark building. The time history variations of important response parameters and evaluation criteria of the benchmark building has been studied for assessing the effectiveness of the isolation systems. The Shape Memory Alloy (SMA) is utilized with elastomeric bearings and friction bearings to study the effectiveness of SMA wires with different isolators. The benchmark building is modelled as a discrete linear elastic shear structure having three degrees of- freedom at each floor level. Time domain dynamic analysis of this building has been carried out with the help of constant average acceleration Newmark’s method and equilibrium of non-linear forces has been taken care by fourth order Runge-Kutta method. The comparative performance of various isolation systems has been studied with uniform and hybrid combinations. The hybrid combination of SMA supplemented bearings works out the better isolation system keeping in view of the percentage reduction in evaluation criteria for smart base-isolated benchmark building. Furthermore, it is shown that, the functionality of SMA wire is not efficient with Lead Rubber Bearing system, as it is able to control displacement but increases the acceleration, base shear, story drift and isolation forces.

Iterative improved reduced system method of fluid-structure interaction with free surface

The aim of this study is to develop an iterative improved reduced system method for the model reduction of the fluid-structure interaction with free surface. The fluid-structure interaction with incompressible fluid for the interior fluid-structure problem involving a free surface without capillarity effects and a linear elastic structure is analyzed. The present study addresses a well-known asymmetric formulation that comprises structural displacement, fluid pressure of the free surface, and fluid pressure of the interior fluid. In this study, the formulation is reduced in a condensation manner, which is degree of freedom based reduced-order modeling. In particular, the proposed iterative scheme considers inertia terms and its equivalent transformation, thereby yielding a reduced model that more accurately reflects the features of the fully coupled system. The performance and convergence of this formulation are evaluated through numerical examples.

Multidisciplinary free-form optimization of solid structures for mean compliance minimization and time-dependent temperature control

A parameter-free free-form optimization method to solve multidisciplinary shape optimization problems of solid structures is presented in this paper. The design objective is both to minimize the mean compliance of linear elastic structures and to control the temperature at arbitrary locations and time periods for arbitrary loadings and under volume constraint, simultaneously. The multidisciplinary design problem is formulated as a distributed-parameter shape optimization problem based on the variational method, and the global objective function is defined using an original normalized weighted sum method. The shape gradient density functions with respect to the shape variation are derived based on the material derivative and the adjoint variable methods. The shape gradient density functions theoretically derived are applied to the H1 gradient method, a gradient method in the function space, to obtain the optimal shape while maintaining the smoothness of the shape. Numerical examples are considered to demonstrate the validity and the effectiveness of the developed optimization method.

Modal control of quasi-static linear-elastic movements of precision equipment

A new modal method for controlling the deformation of a linear elastic structure under conditions of quasi-static loading is proposed. The advantage of the new method is independent control of orthogonal modes, which increases the accuracy and speed of control. To illustrate the proposed method, compensation of the deformation error in a rotary table with a high tilt accuracy is considered. Keywords: modal method, control of structure deformation, quasi-static loading, simulation of a structure, eigenforms adigorev@gmail.com

Reverse shape compensation via a gradient-based moving particle optimization method

Reverse shape compensation is widely used in additive manufacturing to offset the displacement distortion caused by various sources, such as volumetric shrinkage, thermal cooling, etc. Also, reverse shape compensation is also an effective tool for the four-dimensional (4D) printing techniques, shape memory polymers (SMPs), or 3D self-assemble structures to achieve a desired geometry shape under environmental stimuli such as electricity, temperature, gravity etc. In this paper, a gradient-based moving particle optimization method for reverse shape compensation is proposed to achieve a desired geometry shape under a given stimulus. The geometry is described by discrete particles, where the radius basis kernel function is used to realize a mapping from particle to density field, and finite element analysis is used to compute the deformation under the external stimulus. The optimization problem is formulated in detail, and MMA optimizer is implemented to update the location of discrete particles based on sensitivity information. In this work, self-weight due to gravity imposed on linear elastic structures is considered as the source of deformation. The objective of the problem is then to find the initial shape so that the deformed shape under gravity is close to desired geometry shape. A shape interpolation method based on Artificial Neural Network is proposed to reconstruct the accurate geometric prototype. Several numerical examples are demonstrated to verify the effectiveness of proposed method for reverse shape compensation. The computational framework for reverse shape compensation described in this paper has the potential to be extended to consider linear and non-linear deformation induced by other external stimuli.

Modal truncation damping in reduced modal analysis of piecewise linear continuum structures

Modal analysis is an often-used method for the solution of dynamical problems in engineering practice. In the case of continuous systems and discretization methods (e.g., FEM) with a large number of degrees of freedom, the reduced modal analysis is used, where only a selected number of modes are applied, while the higher modes are neglected. A piecewise linear elastic structure has state-dependent stiffness and as such, state-dependent vibration modes. During the dynamic modal analysis of such systems, one has to switch between the modes. During the switch a part of the kinetic and potential energy is transferring into modes, which are neglected in the reduced modal analysis. Thus, using the reduced modal analysis for the numerical calculation of the dynamic behavior of piecewise linear elastic structures introduces the modal truncation damping. In this paper, we present on a continuous system, how this modal truncation damping can be characterized, what are the upper and lower limits of that damping and how the characteristics of the free vibration of a piecewise linear elastic structure can be predicted.

Surface crack and fatigue analysis for cylindrical shaft

In the present work Surface Crack and fatigue analysis for combined cyclic twisting and tension loading using linear elastic fracture mechanics approach and Structure analysis with different orientations of cracks have been performed. The effective stress intensity factor was found to fluctuate during the crack growth when the loading conditions changed, Result shows that when the orientation change from 0 to 10 degree the minimum life of the cylindrical shaft has changed significantly, and its reduced life of the shaft but when the orientation change from 0 to 20 degree its minimum life has been an increase, for the rest 30 degrees and 40 degrees of the crack orientation the minimum life of the shaft get decrease again.

A fast multipole BEM with higher-order elements for 3-D composite materials

The present work is to develop the FMBEM with quadratic elements to the analysis of 3-D linear elastic structures containing subdomains. To the numerical integration, an adaptive method with subdivision of elements is applied. The BEM formulation for perfectly bonded subdomains, in which interface traction forces are eliminated from the system of equations is implemented. The method is applied to problems of solids with spherical inclusions: a single inclusion, two interacting inclusions, many regularly distributed inclusions and randomly distributed inclusions. Different values of volume fraction of inclusions, up to 0.5 for the cubic arrangement of inclusions, and contrast in elastic properties between the matrix and inclusions, up to 1000 for the ratio of Young’s moduli of the constituents, are considered. Stresses or effective elastic constants are computed. The results are compared to analytical models and a good agreement is observed. The proposed method shows a perspective of modelling 3-D composites efficiently.

An efficient and robust method for structural distributed load identification based on mesh superposition approach

Accurate load identification is vital for the structural optimization design and the structural health monitoring in different engineering disciplines. However, the external loads usually are difficult to determine with direct measurements. In the present manuscript, an efficient and robust method is proposed to identify the distributed static or quasi-static loads on linear-elastic structures. The superimposed loading mesh approach is proposed to map the unknown loads, and nodal values of the superimposed loading mesh are taken as inverse variables in the inverse problem. With this strategy, the inverse problem is converted to solving a set of linear equations which maps the relationship between the inverse variables and measured responses. To find the optimal solution, the modified second-order Tikhonov regularization method (TRM) is employed. Furthermore, to ensure the non-negativity of the identified loads, the non-negative least squares (NNLS) is applied. Three numerical tests are employed to verify the feasibility of the proposed method. The results suggest that the distributed loads can be accurately identified, and that the inverse system is well-posed. The proposed method has been found to be robust that reasonable accuracy can be obtained even with significant measurement error.

On periodic homogenization of highly contrasted elastic structures

While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure.

Modal localisation in a beam modelled as a continuous system: A discussion on the use of auxiliary oscillators

In this work, we show that perfectly localised modal vibrations can be obtained in continuous linear elastic structures, provided that, in addition to the regular boundary and continuity/jump conditions, extra (admissibility) conditions are met. They are derived with the aid of: (i) new parameters and/or DOFs of auxiliary systems, and/or (ii) admissible relationships amongst the original structural parameters. In both cases, however, the solution resorts to a fine-tuning problem. To illustrate the main ideas, we discuss three examples: two simply-supported beams with cantilever extensions and attached lumped oscillators, and a “tuning-fork-like” structure. Finally, to show that there are also systems in which fine-tuning is not possible, we discuss the case of two lumped oscillators attached to the free end of a cantilever beam.

Topology optimization of 2D structures with nonlinearities using deep learning

The field of optimal design of linear elastic structures has seen many exciting successes that resulted in new architected materials and structural designs. With the availability of cloud computing, including high-performance computing, machine learning, and simulation, searching for optimal nonlinear structures is now within reach. In this study, we develop convolutional neural network models to predict optimized designs for a given set of boundary conditions, loads, and optimization constraints. We have considered the case of materials with a linear elastic response with and without stress constraint. Also, we have considered the case of materials with a hyperelastic response, where material and geometric nonlinearities are involved. For the nonlinear elastic case, the neo-Hookean model is utilized. For this purpose, we generate datasets composed of the optimized designs paired with the corresponding boundary conditions, loads, and constraints, using a topology optimization framework to train and validate the neural network models. The developed models are capable of accurately predicting the optimized designs without requiring an iterative scheme and with negligible inference computational time. The suggested pipeline can be generalized to other nonlinear mechanics scenarios and design domains.

Unified momentum equation approach for fluid–structure interaction problems involving linear elastic structures

In this article, we present a fully Eulerian monolithic numerical method for simulating fluid–structure interaction problems. This method is based on the unified momentum equation approach, where the elastodynamic equation for linear elastic solids and the incompressible Navier–Stokes equation are unified into a single velocity-based momentum equation that can be solved using a proper numerical method for computational fluid dynamics. The level set method combined with a displacement field extension procedure was employed to capture the moving interfaces. Owing to the formulation scheme applied to the unified momentum equation, both the velocity and stress fields are computed for both the structure and the fluid. Therefore, this method is capable of not only simulating the dynamic interactions of a fluid and structure, but also computing the stress fields in both phases. Three problems were simulated as test examples, a falling disk in a fluid, the oscillation of a flexible rod in a channel, and a bouncing ball, and the results obtained were found to be physically realistic and in good agreement with the published data.

Numerical study on hydroelastic interaction between solitary wave and submerged box

The nonlinear hydroelastic interaction between a solitary wave and a submerged box with elastic upper surface plate is investigated in a 2D numerical wave flume. The objective is to investigate how hydroelasticity affects the interaction process. The numerical wave flume solves incompressible Navier-Stokes equations with a Volume-of-Fluid method for free surface simulation. The linear elastic structure dynamic equations are solved by a Finite Element Method, and embedded into the solution of flow field by using a monolithic fluid-structure interaction algorithm. The numerical method is thoroughly verified against existing experiments and numerical simulations. A series of numerical tests are conducted, which contains free vibrations of the plate, and the interaction between solitary waves and submerged boxes with various wave amplitudes and submerged depths. Results obtained with and without considering hydroelasticity are compared to investigate the influence of hydroelastic effect on the wave load, the structure response and the flow field. Fast Fourier Transform and wavelet transform are adopted to reveal the oscillatory features of the interaction process. Some meaningful conclusions are drawn, such as hydroelasticity changes the amplitude and period of the oscillatory motion and the variation of the oscillatory frequency can be approximated by a simple still water prediction method.

Interval K-L expansion of interval process model for dynamic uncertainty analysis

The interval process model describes a time-variant or dynamic uncertain parameter by the upper and lower bounds rather than the precise probability distribution at each time point, providing an effective structural dynamic uncertainty quantification model with insufficient sample information. By reference to the Karhunen-Loève (K-L) expansion for stochastic process and random field models, a novel expansion method for the interval process model, namely, the interval K-L expansion is proposed in this work. The interval K-L expansion describes the continuous uncertainty of the interval process in time domain by superposition of infinite deterministic time-related functions with uncorrelated interval coefficients, which makes the interval process model much more convenient to use. By introducing a splicing technique, the interval K-L expansion method for multiple correlated interval processes is also presented. Based on the interval K-L expansion, the vibration of structures subject to interval process excitations is analyzed, where the analytic formulation of dynamic response bounds for linear elastic systems and continuum structures are derived. Finally, several numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Pressure influence on elastic wave attenuation in polycrystalline materials.

Traditionally, the acoustoelastic effect refers to the influence of stress in a solid on an elastic wave's phase velocity. Since the phase velocity can be represented by the real part of the complex wave number, a natural question arises regarding the effect of stress on the imaginary part or dissipation of the wave. In this article, the influence of pressure on the elastic wave's attenuation in polycrystalline materials is modeled. The constitutive behavior of an initially stressed solid is coupled into Weaver's scattering-based attenuation model [J. Mech. Phys. Solids 38, 55-86 (1990)]. As a result, the pressure-dependent longitudinal and shear wave attenuation coefficients are unveiled. As the traditional stress-free attenuation coefficients depend on the degree of single-crystal elastic anisotropy, it is shown that the pressure influence on attenuation depends on the anisotropy of the single-crystal's third-order or nonlinear elastic constants. Analysis of the model indicates linkages between pressure derivatives of velocity and attenuation to the material's linear and nonlinear elastic anisotropy, crystal structure, and type of atomic bonding.

CANONICAL DUAL FINITE ELEMENT METHOD FOR SOLVING NONCONVEX MECHANICS AND TOPOLOGY OPTIMISATION PROBLEMS

Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly…

Robust topology optimization for multi-material structures under interval uncertainty

In this paper, we propose an efficient method to design robust multi-material structures under interval loading uncertainty. The objective of this study is to minimize the structural compliance of linear elastic structures. First, the loading uncertainty can be decomposed into two unit forces in the horizontal and vertical directions based on the orthogonal decomposition, which separates the uncertainty into the calculation coefficients of structural compliance that are not related to the finite element analysis. In this manner, the time-consuming procedure, namely, the nested double-loop optimization, can be avoided. Second, the uncertainty problem can be transformed into an augmented deterministic problem by means of uniform sampling, which exploits the coefficients related to interval variables. Finally, an efficient sensitivity analysis method is explicitly developed. Thus, the robust topology optimization (RTO) problem considering interval uncertainty can be solved by combining orthogonal decomposition with uniform sampling (ODUS). In order to eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, the relative uncertainty related to interval objective function is employed to characterize the structural robustness. Several multi-material structure optimization cases are provided to demonstrate the feasibility and efficiency of the proposed method, where the magnitude uncertainty, directional uncertainty, and combined uncertainty are investigated.