Linear Elastic Structures Research Articles

Stress Field Prediction in Cantilevered Structures Using Convolutional Neural Networks

Abstract The demand for fast and accurate structural analysis is becoming increasingly more prevalent with the advance of generative design and topology optimization technologies. As one step toward accelerating structural analysis, this work explores a deep learning-based approach for predicting the stress fields in 2D linear elastic cantilevered structures subjected to external static loads at its free end using convolutional neural networks (CNNs). Two different architectures are implemented that take as input the structure geometry, external loads, and displacement boundary conditions, and output the predicted von Mises stress field. The first is a single input channel network called SCSNet as the baseline architecture, and the second is the multichannel input network called StressNet. Accuracy analysis shows that StressNet results in significantly lower prediction errors than SCSNet on three loss functions, with a mean relative error of 2.04% for testing. These results suggest that deep learning models may offer a promising alternative to classical methods in structural design and topology optimization. Code and dataset are available.2

On the solutions to the Saint–Venant problem of heterogeneous beam-like structures with periodic microstructures

This paper presents displacement solutions of the Saint-Venant problem for linear elastic heterogeneous beam structures with arbitrarily shaped periodic microstructures. The solutions are generated by introducing unknown axially periodic displacement functions in place of the axially invariant warping functions in Ieşan's rational scheme, due to the periodic characteristics of the beam structure, while retaining the displacement fields integrated from rigid-body displacements, which represent basic kinematics in beam theories. The governing equations of the unknown functions, defined on a base cell, are then developed from equilibrium equations and boundary conditions of the periodic beam structure. Moreover, an improved FE formulation of the governing equations is presented and its efficient numerical implementation approach, which can be readily realized using FE software as a black box, is proposed with the aid of NIAH (Novel Numerical Implementation of Asymptotic homogenization) approach. To illustrate the validity of the Saint-Venant solution and the effectiveness of the numerical implementation approach, several numerical examples are presented and compared with detailed three-dimensional FE analysis for the heterogeneous beam structures in terms of stress components.

Topology optimization for design-dependent hydrostatic pressure loading via the level-set method

A few level-set topology optimization (LSTO) methods have been proposed to address complex fluid-structure interaction. Most of them did not explore benchmark fluid pressure loading problems and some of their solutions are inconsistent with those obtained via density-based and binary topology optimization methods. This paper presents a LSTO strategy for design-dependent pressure. It employs a fluid field governed by Laplace’s equation to compute hydrostatic fluid pressure fields that are loading linear elastic structures. Compliance minimization of these structures is carried out considering the design-dependency of the pressure load with moving boundaries. The Ersatz material approach with fixed grid is applied together with work equivalent load integration. Shape sensitivities are used. Numerical results show smooth convergence and good agreement with the solutions obtained by other topology optimization methods.

OpenFPCI: A parallel fluid–structure interaction framework

This paper presents OpenFPCI, a framework for coupling the C++ toolbox OpenFOAM-Extend, a computational fluid dynamics package, with the general purpose finite element package ParaFEM, written in Fortran and used to solve structural mechanics problems. The coupling of these two open source and scalable toolboxes, facilitates the use of high performance computing resources for the solution of fluid–structure interaction problems. The framework uses a master–slave approach, with OpenFOAM-Extend acting as the master and calling OpenFPCI plugins. The plugins are composed of a series of subroutines used to initialise and solve a specific engineering problem and make use of ParaFEM’s highly parallel implementation. The plugins are wrapped by C constructs such that OpenFOAM-Extend can call these Fortran subroutines consistently and when the solution from ParaFEM is required. Each plugin solves a different solid mechanics problem, with the current features including the deformation of a linear-elastic structure undergoing small strain and the deformation of a St. Venant–Kirchhoff material. Throughout this paper the focus will lie on the large strain plugin, considering the implementation and its validation for a benchmark problem, along with assessment of parallel capabilities, which are shown to scale to three thousand cores. This paper will be of interest to OpenFOAM and ParaFEM practitioners looking to utilise multiphysics simulations for their research, along with researchers looking to integrate fluid–structure interaction into their studies. Program summaryProgram Title:OpenFPCIProgram Files doi:http://dx.doi.org/10.17632/ntprzxk477.1Licensing provisions: BSD 2-ClauseProgramming language: Fortran, C and C++External libraries:OpenFOAM and ParaFEMSupplementary material: Example test cases are available within the OpenFPCI repository.Nature of problem: OpenFPCI was developed to solve computationally expensive fluid structure interaction problems by running on high performance computing systems. The framework was designed to enable the coupling of advanced ParaFEM capabilities to OpenFOAM-Extend.Solution method: OpenFOAM-Extend uses the classic Arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations to deal with moving boundaries. The moving boundary is defined by using an OpenFPCI plugin, using ParaFEM’s libraries, to solve the deformation of the adjoining structure.

Analysis of 3D linear elastic masonry-like structures through the API of a finite element software

A new numerical procedure is presented to perform the analysis of three-dimensional linear elastic no-tension structures exploiting the application programming interface of a general purpose finite element analysis software. Masonry is replaced by an equivalent orthotropic material with spatially varying elastic properties and negligible stiffness in the case of cracking strain. A non-incremental algorithm is implemented to define the distribution of the equivalent material, minimizing the strain energy so as to achieve a compression-only state of stress for any given compatible load. Applications are shown for masonry-like solids of general shape visualizing load paths in walls subject to dead loads and out-of-plane live loads, circular domes under self-weight and a groin vault acted upon by both vertical and horizontal seismic loading.

Optimal fuzzy iterative learning control based on artificial bee colony for vibration control of piezoelectric smart structures

Combining P-type iterative learning (IL) control, fuzzy logic control and artificial bee colony (ABC) algorithm, a new optimal fuzzy IL controller is designed for active vibration control of piezoelectric smart structures. In order to accelerate the learning speed of feedback gain, the fuzzy logic controller is integrated into the ANSYS finite element (FE) models by using APDL (ANSYS Parameter Design Language) approach to adjust adaptively the learning gain of P-type IL control. For improving the performance and robustness of the fuzzy logic controller as well as diminishing human intervention in the operation process, ABC algorithm is used to automatically identify the optimal configurations for values in fuzzy query table, fuzzification parameters and defuzzification parameters, and the main program of ABC algorithm is operated in MATLAB. The active vibration equations are driven from the FE equations for the dynamic response of a linear elastic piezoelectric smart structure. Considering the vibrations generated by various external disturbances, the optimal fuzzy IL controller is numerically investigated for a clamped piezoelectric smart plate. Results demonstrate that the proposed control approach makes the feedback gain has a fast learning speed and performs excellent in vibration suppression. This is demonstrated in the results by comparing the new control approach with the P-type IL control.

Cavity identification in elastic structures by explicit domain mapping and boundary mode sensitivity analysis

This paper presents a novel cavity identification approach for linear elastic structures with the boundary mode data. The proposed approach consists of two major aspects. The first is the construction of an explicit domain mapping that maps the reference domain onto the physical domain by the level set functions. In proceeding so, the R-functions theory not only provide an analytical methodology to get the level set functions for the external and interior cavity boundaries, but also construct the desired domain mapping directly by level-set-functions weighted interpolation of the parameterized boundary mappings. The physical domain with the cavities is explicitly parameterized in the domain mapping and the desired mesh keeps fixed over the reference domain. The second is the enhanced sensitivity-based approach for minimum solution to the nonlinear least-squares goal function of the cavity identification problem. The sensitivity of the boundary modes to the geometric parameters is analyzed with the help of the acquired domain mapping and the trust-region constraint is introduced in order to reach a weak convergence. Numerical examples are conducted to show the feasibility and efficiency of the present cavity identification approach.

Free-form optimization method for controlling time-dependent response of 3D linear elastic structure

Free-form optimization method for controlling time-dependent response of 3D linear elastic structure

Modeling, identification, and optimization of violin bridges

AbstractWe present the derivation and implementation of a mathematical model for the linear elastic structure of a violin bridge. In favour of getting highly accurate simulation results, special effort has been taken to reconstruct the computational geometry from a µ‐CT scan. After laying out our approach for this in detail, we also present first vibroacoustical analysis like solving the eigenvalue problem of elasticity for the orthotropic material law of the wooden bridge and simulation of its dynamical behaviour. We then conclude with explaining our ongoing work of identifiying material paramters by an inverse problem. With this project we are aiming to optimize a violin bridge regarding its material parameters and its shape.

An interval finite element method for the analysis of structures with spatially varying uncertainties

Finite element analysis of linear-elastic structures with spatially varying uncertain properties is addressed within the framework of the interval model of uncertainty. Resorting to a recently proposed interval field model, the uncertain properties are expressed as the superposition of deterministic basis functions weighted by particular unitary intervals. An Interval Finite Element Method (IFEM) incorporating the interval field representation of uncertainties is formulated by applying an interval extension in conjunction with the standard energy approach. Uncertainty propagation analysis is performed by adopting a response surface approach which provides approximate explicit expressions of response bounds requiring only a few deterministic analyses. Then, the whole procedure is implemented in ABAQUS’ environment by coding User Subroutines and Python scripts.2D plane stress and bending problems involving uncertain Young's modulus of the material are analyzed. The accuracy of the proposed IFEM as well as response variability under spatially dependent uncertainty are investigated.

Bayesian sparse regularization for multiple force identification and location in time domain

ABSTRACTIn this work, reconstruction and location in time domain of multiple forces acting on a linear elastic structure are achieved through a Bayesian approach to solve an inverse problem. The Bayesian solution of the inverse problem is provided in the form of a posterior probability density function. The unknown forces are determined through Markov chain Monte Carlo method, the Gibbs algorithm. This posterior density integrating both the likelihood and prior information was considered for the particular case of a linear elastic structure. The measurements are affected by an additive random noise. Two particular cases were analysed: unperturbed and uncertain model representing the structure. The unperturbed model was used to identify a single force. When the model is uncertain, compressed sensing technique was used to provide an adequate sparse representation of the inverse problem through a wavelet basis. With this sparse representation, the possibility of achieving automatic location of the forces was investigated. This requires to identify all the degrees of freedom along with the identified actions are not vanishing. Also, the possibility of force identification with less sensors than forces was studied. The proposed approach is illustrated and validated on numerical examples. This proposed approach is compared with classical approach of force identification based on Tikhonov regularization associated with the GCV criterion.

Low-pass-filter-based shock response spectrum and the evaluation method of transmissibility between equipment and sensitive components interfaces

According to the features of the sources of pyroshock and ballistic shock, this study considers the pyroshock and ballistic shock generated by their respective impulsive sources as damped harmonic waves with different frequencies. According to the linear superposition assumption of damped harmonic waves in a linear elastic structure, a shock analysis method based on low-pass-filtered shock signals and their corresponding shock response spectrum (SRS), termed as low-pass-filter-based shock response spectrum (LPSRS), is proposed. LPSRS contains rich information of the frequency distribution of the shock excitation signal. A method to calculate shock transmissibility is proposed based on LPSRS and basic modal information of the equipment structure. LPSRS and SRS curves can be predicted at any given position of the equipment structure. The prediction method is validated by finite element method (FEM) simulation.

An Efficient Topology Description Function Method Based on Modified Sigmoid Function

Optimal geometries extracted from traditional element-based topology optimization outcomes usually have zigzag boundaries, leading to being difficult to fabricate. In this study, a fairly accurate and efficient topology description function method (TDFM) for topology optimization of linear elastic structures is developed. By employing the modified sigmoid function, a simple yet efficient strategy is presented to tackle the computational difficulties because of the nonsmoothness of Heaviside function in topology optimization problem. The optimization problem is to minimize the structural compliance, with highest stiffness, while satisfying the volume constraint. The design problem is solved by a Sequential Linear Programming method. Convergent, crisp, and smooth final layouts are obtained, which can be fabricated without postprocessing, demonstrated by a series of numerical examples. Further, the proposed method has a rather higher accuracy and efficiency compared with traditional TDFM, when the classical topology optimization methods, such as bidirectional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP) method, are taken as benchmark.

Evaluation of the FMBEM efficiency in the analysis of porous structures

PurposeThe purpose of this paper is to evaluate the efficiency of the fast multipole boundary element method (FMBEM) in the analysis of stress and effective properties of 3D linear elastic structures with cavities. In particular, a comparison between the FMBEM and the finite element method (FEM) is performed in terms of accuracy, model size and computation time.Design/methodology/approachThe developed FMBEM uses eight-node Serendipity boundary elements with numerical integration based on the adaptive subdivision of elements. Multipole and local expansions and translations involve solid harmonics. The proposed model is used to analyse a solid body with two interacting spherical cavities, and to predict the homogenized response of a porous material under linear displacement boundary condition. The FEM results are generated in commercial codes Ansys and MSC Patran/Nastran, and the results are compared in terms of accuracy, model size and execution time. Analytical solutions available in the literature are also considered.FindingsFMBEM and FEM approximate the geometry with similar accuracy and provide similar results. However, FMBEM requires a model size that is smaller by an order of magnitude in terms of the number of degrees of freedom. The problems under consideration can be solved by using FMBEM within the time comparable to the FEM with an iterative solver.Research limitations/implicationsThe present results are limited to linear elasticity.Originality/valueThis work is a step towards a comprehensive efficiency evaluation of the FMBEM applied to selected problems of micromechanics, by comparison with the commercial FEM codes.

Dynamics of structures with uncertain-but-bounded parameters via pseudo-static sensitivity analysis

Abstract This paper deals with the analysis of linear-elastic structures with uncertain-but-bounded parameters subjected to deterministic dynamic loads. A novel procedure based on the use of sensitivity analysis in conjunction with classical modal superposition method is proposed. Specifically, a pseudo-static sensitivity analysis is performed to seek the combinations of the endpoints of the uncertain parameters which give the lower bound and upper bound of the response at each time instant. Among these, two combinations are selected as the most common ones over the time interval of interest in order to avoid the onerous updating of the uncertain parameters at each time instant. Then, the bounds of the response time-history are evaluated by performing two parallel deterministic modal analyses associated to the most common combinations of the extreme values of interval uncertainties. Numerical results demonstrate that the proposed method is more efficient than both the Interval Perturbation Method (IPM) and the classical combinatorial procedure. Furthermore, unlike the IPM, it allows the analysis of large-size structures exhibiting relatively large fluctuations of the uncertain parameters.

Numerical analysis of wave-induced poro-elastic seabed response around a hexagonal gravity-based offshore foundation

In order to prevent the future risk of soil and structural failures for the offshore foundations, it is essential to evaluate the seabed soil behaviors in the vicinity of the foundation under dynamic wave loadings. The objective of this paper is to investigate the wave-induced soil response and liquefaction risk around a hexagonal gravity-based offshore foundation. Three-dimensional (3D) numerical analysis is performed by applying an integrated multiphysics model developed in the finite volume method (FVM) based OpenFOAM framework. The integrated model incorporates solvers of the nonlinear waves, the linear elastic structure and the anisotropic poro-elastic seabed soil. The free surface model and soil model are verified by grid convergence studies. The wave-induced soil response model is validated by reproducing a laboratory experiment and a good agreement is obtained.Distributions of wave-induced shear stress, pore pressure, vertical displacement and seepage flow structure in the seabed are investigated. It is found that the presence of the foundation significantly amplifies the wave-induced shearing effect and vertical displacement in the underlying seabed soil. Seabed consolidation state in the presence of the structure is evaluated. Since the foundation is embedded in the seabed at a depth, the vertices of the hexagonal foundation cause the stress concentration in the nearby soil during the consolidation process. Therefore, the momentary liquefaction at the vertices is not as significant as that at the edges due to the high initial effective stress. A parametric study with different wave heights is conducted to examine the changes of soil response and momentary liquefaction depth around the hexagonal foundation. Effects of isotropic and anisotropic soil permeability on the pore pressure distribution are investigated. It shows that the effect of anisotropic permeability should be considered for the medium sand that is modelled in the present study.

An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions

This paper presents a compact and efficient 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (RBFs), which is applied to minimize the compliance of a two-dimensional linear elastic structure. This parameterized level set method using radial basis functions can maintain a relatively smooth level set function with an approximate re-initialization scheme during the optimization process. It also has less dependency on initial designs due to its capability in nucleation of new holes inside the material domain. The MATLAB code and simple modifications are explained in detail with numerical examples. The 88-line code included in the appendix is intended for educational purposes.

Elastic–Viscoelastic Composite Structures Analysis With an Improved Burgers Model

Composite structures integrated with viscoelastic materials are becoming more and more popular in the application of vibration suppression. This paper presents a comprehensive approach for analyzing this class of structures with an improved Burgers model, from material constitutive modeling, finite element formulation to solution method. The refined model consists of a spring component and multiple classical Burgers components in parallel, where the spring component converts the viscoelastic fluid model to a viscoelastic solid model and the multiple Burgers components increase the accuracy. Through the introduction of auxiliary coordinates, the model is applied to the finite element formulation of composites structures with viscoelastic materials. Consequently, a complicated Volterra integro-differential equation is transformed into a standard second-order differential equation and solution techniques for linear elastic structures can be directly used for elastic–viscoelastic composite structures. The improved Burgers model is a second-order mini-oscillator model, in which every mini-oscillator term has four parameters. The model parameters determination is performed by optimization algorithm. By comparison of model fitting results for a typical viscoelastic material, the refined model is better in accuracy than Golla–Hughes–McTavish (GHM) model and original Burgers model. Finally, several numerical examples are presented to further verify the effectiveness of the improved Burgers model.

Fluid‐structure coupling of linear elastic model with compressible flow models

SummaryCavitation erosion is caused in solids exposed to strong pressure waves developing in an adjacent fluid field. The knowledge of the transient distribution of stresses in the solid is important to understand the cause of damaging by comparisons with breaking points of the material. The modeling of this problem requires the coupling of the models for the fluid and the solid. For this purpose, we use a strategy based on the solution of coupled Riemann problems that has been originally developed for the coupling of 2 fluids. This concept is exemplified for the coupling of a linear elastic structure with an ideal gas. The coupling procedure relies on the solution of a nonlinear equation. Existence and uniqueness of the solution is proven. The coupling conditions are validated by means of quasi‐1D problems for which an explicit solution can be determined. For a more realistic scenario, a 2D application is considered where in a compressible single fluid, a hot gas bubble at low pressure collapses in a cold gas at high pressure near an adjacent structure.

Impedance approach to designing efficient vibration energy absorbers

The concept introduced previously by the authors on the best sound absorber having the maximum allowable efficiency in absorbing the energy of an incident sound field has been extended to arbitrary linear elastic media and structures. Analytic relations have been found for the input impedance characteristics that the best vibrational energy absorber should have. The implementation of these relations is the basis of the proposed impedance method of designing efficient vibration and noise absorbers. We present the results of a laboratory experiment that confirms the validity of the obtained theoretical relations, and we construct the simplest best vibration absorber. We also calculate the parameters and demonstrate the efficiency of a dynamic vibration absorber as the best absorber.