Linear Elastic Structures Research Articles

Stress-based topology optimization using maximum entropy basis functions-based meshless method

AbstractThis paper presents volume-constrained stress minimization-based, topology optimization. The maximum entropy (maxent) basis functions-based meshless method for two-dimensional linear elastic structures is explored. This work focuses to test the effectiveness of the meshless method in handling the stress singularities during the topology optimization process. The commonly used moving least square basis functions are replaced with maximum entropy basis functions, as the latter possess weak Kronecker delta property which leads to the finite element method (FEM) like displacement boundary conditions imposition. The maxent basis functions are calculated once at the beginning of the simulation and then used in optimization at every iteration. Young’s modulus for each background cell is interpolated using the modified solid isotropic material with penalization approach. An open source pre-processor CUBIT is used. A comparison of the proposed approach with the FEM is carried out using a diverse set of problems with simple and complex geometries of structured and unstructured discretization, to establish that maxent-based meshless methods perform better in tackling the stress singularities due to its smooth stress field.

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

Multi-scale design of composite material structures for maximizing fundamental natural frequency

This study deals with the topology optimization (TO) of linear elastic composite structures considering fundamental natural frequency maximization. The concept of functionally graded structures (FGS) is used for anisotropic composite material structure design to improve the structural dynamic performance. The rational approximation of material properties (RAMP) method was employed as an alternative to the solid isotropic material with penalization (SIMP) method to improve the convergence stability for eigen-frequency problem. In the SIMP method, the occurrence of local eigen-modes at low frequencies in the void region causes ill-convergence, whereas the RAMP method prevents local eigen-modes from occurring. The method of moving asymptotes is used as an optimizer. Numerical verification was performed for three structural types: cantilever beam, double-clamped beam, and four-pinned beam structure. In addition, for cantilever beam design, bi-objective optimization for maximizing fundamental natural frequency as well as minimizing structural compliance was performed using the adaptive weighting method that quantitatively reflects the designer's preference by adjusting the weight for each objective function at each iteration.

An analytical model for dynamic analysis of inhomogeneous sandwich plate

Sandwich plates are widely used in the engineering domain and are extensively studied by researchers for their special performances. Among them, the sandwich plate with coordinate-dependent material parameters has excellent vibration isolation performance. However, there are few effective and convenient analytical methods for solving the dynamic response of this type of sandwich plate. To solve this problem, an equivalent spring-based model for the dynamic analysis of the sandwich plate is proposed in this paper. Assuming that the sandwich plate is a linear elastic structure, and considering the core plate as equivalent springs, whose stiffness coefficients are derived based on the classical elasticity theory and Kirchhoff hypothesis. Besides, displacement solutions of base and constrained plates of the sandwich plate in the x, y and z directions are expressed as a two-dimensional Fourier series supplemented with several one-dimensional Fourier series, respectively. Then, according to the orthogonality of trigonometric functions and selectivity of Dirac functions, the displacements of the sandwich plate are solved accurately. By comparing with available analytical and numerical results, the proposed method is proven to have excellent performance in dealing with the sandwich plate with coordinate-dependent material parameters. Finally, the dynamic characteristics and vibration isolation performance of the inhomogeneous sandwich plate are discussed. The results show that the proposed method can be used to design the inhomogeneous sandwich plate and improve their vibration isolation performance significantly.

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

An energy approach to asymptotic, higher-order, linear homogenization

A higher-order homogenization method for linear elastic structures is proposed. While most existing approaches to homogenization start from the equations of equilibrium, the proposed one works at the energy level. We start from an energy functional depending on microscopic degrees of freedom on the one hand and on macroscopic variables on the other hand; the homogenized energy functional is derived by relaxing the microscopic degrees of freedom and applying a formal two-scale expansion. This method delivers the the energy functional of the homogenized model directly, including boundary terms that have not been discussed in previous work. Our method is formulated in a generic setting which makes it applicable to a variety of geometries in dimension 1, 2 or 3, and without any particular assumption on material symmetry. An implementation using a symbolic calculation language is proposed and it is distributed as an open-source library. Simple illustrations to elastic trusses having pre-stress or graded elastic properties are presented. The approach is presented in the context of discrete elastic structures and the connection with previous work on the higher-order homogenization of period continua is discussed.

A compatible boundary condition-based topology optimization paradigm for static mechanical cloak design

We propose a novel compatible boundary condition-based topology optimization formulation for realizing static mechanical cloaking of linear elastic structures. The key idea is to make the actual boundary conditions along the exterior of the cloaking region consistent with the reference boundary conditions, including the displacement and the nodal force, by designing the cloak structure solely. This compatible boundary condition-based optimization paradigm is not only applicable to various shaped voids (e.g., circle, ellipse, star, Mickey, square, “DUT” logo) and diverse loading conditions (e.g., pressure-sliding, biaxial-stretching, linear-stretching, random-stretching, stretching-sliding) but also demonstrates insensitivity to element quantities and slight variations of external load, ensuring consistent and robust elastostatic cloaking performance. Additionally, adopting the Moving Morphable Voids (MMV) method guarantees the optimized structures have the merits of explicit geometric boundaries and easy fabrication. Numerical examples and experimental validation demonstrate the effectiveness of the proposed approach. The present approach can be easily extended for cloak design in three dimensions and multiphysics.

3D numerical investigation of wave-induced seabed response around a tripod pile foundation with different arrangements

This paper developed an integrated numerical model for the dynamic response of the seabed around tripod turbine foundations under dynamic wave pressure. Three-dimensional (3D) numerical analysis is performed by applying an integrated multiphysics model developed in the finite volume method (FVM) based on the OpenFOAM framework. The present work adopts nonlinear wave model, linear elastic structure model, and anisotropic porous seabed model for wave-structure-seabed interactions. Accuracy of the wave-structure-seabed coupling process was ensured by comparing the numerical model with the numerical and experimental results. This study indicates: (1) the existence of tripods will significantly affect the consolidation results of the surrounding seabed; (2) wave force acting on the tripod structure varies with different arrangements, in which the maximum wave force under the wave crest decreases as the angle between wave and structure increases; and (3) the asymmetry 15° arrangement would significantly increase the maximum liquefaction depth of the seabed.

Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction

AbstractIn this article, a methodology for fine scale modeling of large scale linear elastic structures is proposed, which combines the variational multiscale method, domain decomposition and model order reduction. The influence of the fine scale on the coarse scale is modeled by the use of an additive split of the displacement field, addressing applications without a clear scale separation. Local reduced spaces are constructed by solving an oversampling problem with random boundary conditions. Herein, we inform the boundary conditions by a global reduced problem and compare our approach using physically meaningful correlated samples with existing approaches using uncorrelated samples. The local spaces are designed such that the local contribution of each subdomain can be coupled in a conforming way, which also preserves the sparsity pattern of standard finite element assembly procedures. Several numerical experiments show the accuracy and efficiency of the method, as well as its potential to reduce the size of the local spaces and the number of training samples compared to the uncorrelated sampling.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

On the identification of Lamé parameters in linear isotropic elasticity via a weighted self-guided TV-regularization method

Abstract Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV \mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.

The role of soil in structure response of a building damaged by the 26 December 2018 earthquake in Italy

Local soil conditions can significantly modify the seismic motion expected on the soil surface. In most cases, the indications concerning the influence of the underlying soil provided by the in-force European and Italian Building Codes underestimate the real seismic amplification effects. For this reason, numerical analyses of the local seismic response (LSR) have been encouraged to estimate the soil filtering effects. These analyses are generally performed in free-field conditions, ignoring the presence of superstructures and, therefore, the effects of dynamic soil-structure interaction (DSSI). Moreover, many studies on DSSI are characterised by a sophisticated modelling of the structure and an approximate modelling of the soil (using springs and dashpots at the foundation level); while others are characterised by a sophisticated modelling of the soil and an approximate modelling of the structure (considered as a simple linear elastic structure or a single degree of freedom system). This paper presents a set of finite element method (FEM) analyses on a fully-coupled soil-structure system for a reinforced concrete building located in Fleri (Catania, Italy). The building, designed for gravity loads only, was severely damaged during the 26 December 2018 earthquake. The soil was modelled considering an equivalent visco-elastic behaviour, while the structure was modelled assuming both the visco-elastic and visco-inelastic behaviours. The comparison made between the results of the FEM analyses and the observed damage is valuable. • Soil filtering phenomena are a key aspect for structure seismic response. • FEM modelling of the soil cannot ignore the dynamic properties of the bedrock. • Modelling of the structures should be performed in the inelastic range. • Earthquake damage is not necessarily a "physical" process for vulnerable buildings. • FEM analyses of DSSI must be encouraged for all buildings.

Evaluating dynamic behaviour of a concrete dam using modal analysis

Modal Analysis is the most fundamental aspect of the dynamic study of structures. It is used to determine the vibrational characteristics of linear elastic structures. It suggests the natural frequencies and mode shapes, which are essential to understand the dynamic behaviour of that structure. In this study, Shahpurkandi Dam, a concrete gravity dam of height 55.5 m is considered in order to investigate its dynamic properties using the finite element package ‘ANSYS’. Furthermore, structure-acoustic interaction is explicitly modelled, and a case of the full reservoir is taken into account to analyze the modal response of the dam. The first 6 mode shapes are extracted, and the total displacement is recorded through the numerical simulation of the direct solver including the block Lanczos method. This study shows the maximum deformation is recorded for the first fundamental natural frequency, which is 0.0039 mm at the crest of the dam. Henceforth, it is essential to monitor the mode shape and natural frequency of initial modes. The comparison of natural frequency is also demonstrated for full reservoir and empty reservoir cases.

Seismic Performance of Irregular Building with different Variable sliding isolators and Semi active Dampers

The comparative performances of semiactive friction and stiffness dampers with different control laws in the base-isolated Irregular building subjected to bi-directionally acting strong earthquakes have been studied. The Irregular building is hybridly isolated with rubber bearings and friction pendulum system (FPS) or variable frequency pendulum isolator (VFPI) or variable curvature friction pendulum system (VCFPS). The shear type base-isolated Irregular building is modeled as three-dimensional linear elastic structure having three degrees-of-freedom at each floor level. Time domain dynamic analysis of the building has been carried out with the help of constant average acceleration Newmark-Beta method and non-linear isolation forces has been taken care by fourth-order Runge-Kutta method. The effects of variation of characteristic properties of semiactive dampers on their hysteresis loops and on the structural response of Irregular building is studied through parametric study. Comparative performances of different semiactive dampers with sliding isolation systems for seismic control of Irregular building have been observed through time history plots and peak response performance indices. It has been found that semiactive electromagnetic friction damper work efficiently with VFPI and VCFPS in comparison FPS for the Irregular building for near field earthquakes as it not only reduces base displacement at lower control force but also give lower base shear and story drift in base-isolated Irregular building. The control laws based on modulated homogeneous friction control semiactive friction dampers better than the predictive control law.

Application of MR damper in Base-isolated Irregular Building with different variable sliding Isolators

Application of Electromagnetic induction (EMI) based passively controlled and some local response governed MR dampers in controlling the isolator displacement in base-isolated Irregular building have been studied and compared with the Lyapunov controller. The Irregular building is hybrid isolated with elastomeric rubber bearings and a variety of sliding isolators subjected to strong near field earthquakes acting bi-directionally in horizontal plane. The shear type base-isolated Irregular building is modelled as three-dimensional linear elastic structure having three degrees-of-freedom at each floor level. Time domain dynamic analysis of the building has been carried out using constant-average acceleration Newmark-Beta method and non-linear isolation forces has been taken care by fourth-order Runge-Kutta method. It is observed that the EMI based passive controller performs better than the Lyapunov controller though it reduces less base displacement but gives lower structural response among all the controllers. Local response governed Hyperbolic and Aly’s controller require only velocity and displacement at controller locations sound simpler in application as compared to Lyapunov controller especially to civil engineering structures like Irregular building. The Hyperbolic controller restrains base displacement comparable to the Lyapunov controller but gives less base shear and story drifts to Irregular building. Variable frequency pendulum isolator (VFPI) and variable curvature friction pendulum system (VCFPS) sliding isolators perform better with MR damper control in comparison to conventional friction pendulum system (FPS) in base–isolated Irregular building.

A fidelity equivalence computation method for topology optimization of geometrically nonlinear structures

The application of geometrically nonlinear topology optimization under large deformations in engineering is seriously hindered by its excessive time consumption. To this end, a fidelity equivalent computation method (FECM) with double optimization loops is proposed, in which the topology optimization (TO) model of linear elastic structure is regarded as the low-fidelity TO model of a geometrically nonlinear structure. Then, the geometrically nonlinear TO model is approached in a piecewise manner by a linear TO model using the equivalent factor. Accordingly, the sensitivities of geometrically nonlinear TO are transformed into the sensitivities of linear TO to capture the layouts of the geometrically nonlinear TO model. Moreover, computational accuracy is demonstrated. The validity and feasibility of the proposed FECM are verified through five numerical examples, and the results demonstrate that the FECM can provide sufficient accuracy for solving the TO problem of geometrically nonlinear structures with a relatively small computational cost.

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Proportional topology optimisation with maximum entropy-based meshless method for minimum compliance and stress constrained problems

In this paper, proportional topology optimisation (PTO) with maximum entropy (maxent)-based meshless method is presented for two-dimensional linear elastic structures for both minimum compliance (PTOc) and stress constraint (PTOs) problems. The computation of maxent basis functions is efficient as compared to the standard moving least square (MLS) and possesses a weak Kronecker delta property leading to straightforward imposition of Dirichlet boundary conditions. The PTO is a simple, non-gradient, accurate, and efficient method compared to the standard topology optimisation methods. A detailed and efficient implementation of the computational algorithms for both PTOc and PTOs is presented. The maxent basis functions are calculated only once at the start of simulation and used in each optimisation iteration. Young’s modulus for each background cells is calculated using the modified solid isotropic material with penalisation (SIMP) method. A parametric study is also conducted on the degree of proportionality and history dependence of both PTOc and PTOs algorithms. A variety of numerical examples with simple and complex geometries, and structured and unstructured discretisations are presented to show the accuracy, efficiency, and robustness of the developed computational algorithms. Both PTOc and PTOs algorithms can handle large topological changes, and provide excellent optimisation convergence characteristics.

A priori error analysis of shape derivatives of linear functionals in structural topology optimization

We provide theoretical analyses and numerical comparisons of boundary-based and volumetric shape derivative expressions of linear objective functionals encountered in topology optimization of linear elastic structures. The two expressions yield identical results if the domain is smooth and the governing equation is solved exactly; however, the finite element approximation of the expressions for less regular domains yield different results. We first review the two expressions to show that the volumetric shape derivative places weaker regularity requirements, which, unlike the requirements for boundary-based shape derivatives, are satisfied in most finite element approximations. We then analyze the error in the degree-k polynomial finite element approximations of the two expressions; we show that, for sufficiently regular problems, the boundary-based and volumetric shape derivatives provide kth and 2k-th order accurate approximations, respectively, of the true shape derivative. We finally assess, through numerical examples, the practical implications of using the volumetric vs boundary-based shape derivatives in topology optimization problems; we demonstrate that methods based on the volumetric shape derivative yield more robust solutions to topology optimization problems.

Probability distribution estimation for harmonisable loads and responses of linear elastic structures

In probabilistic distribution-based structural dynamics representing joint probability density functions (JPDFs) of loads and those of structural responses at any collection of time instants is a challenging and important issue. In this study, based on the mean square (m.s.) stochastic calculus, a novel method for estimating the JPDFs of external loads and those of structural responses is proposed. In particular, an external load is modelled as a harmonisable process, and its two approximate representations are proposed by the discrete Fourier transform. The JPDFs of the Fourier coefficients are directly estimated from multiple load samples. Using the load JPDFs in the frequency domain can estimate the JPDFs of the load at any finite time instants. By applying the load representations to the dynamic governing equation of linear elastic structures, the JPDFs of the structural responses at any finite time instants can be estimated. The two load representations are proved to converge in m.s. to the exact load process under certain conditions. Correspondingly, the estimated JPDFs of the load and those of structural responses in the time domain converge to their corresponding exact ones. The proposed method is numerically verified using a linear elastic structure under a nonstationary earthquake ground motion.