Approximation Of Displacement Field Research Articles

DMF-TONN: Direct Mesh-free Topology Optimization using Neural Networks

AbstractWe propose a direct mesh-free method for performing topology optimization by integrating a density field approximation neural network with a displacement field approximation neural network. We show that this direct integration approach can give comparable results to conventional topology optimization techniques, with an added advantage of enabling seamless integration with post-processing software, and a potential of topology optimization with objectives where meshing and Finite Element Analysis (FEA) may be expensive or not suitable. Our approach (DMF-TONN) takes in as inputs the boundary conditions and domain coordinates and finds the optimum density field for minimizing the loss function of compliance and volume fraction constraint violation. The mesh-free nature is enabled by a physics-informed displacement field approximation neural network to solve the linear elasticity partial differential equation and replace the FEA conventionally used for calculating the compliance. We show that using a suitable Fourier Features neural network architecture and hyperparameters, the density field approximation neural network can learn the weights to represent the optimal density field for the given domain and boundary conditions, by directly backpropagating the loss gradient through the displacement field approximation neural network, and unlike prior work there is no requirement of a sensitivity filter, optimality criterion method, or a separate training of density network in each topology optimization iteration.

Nonlinear Analysis of Isotropic Rectangular Thin Plates Using Ritz Method

This research work presents Nonlinear Analysis of Isotropic Thin Rectangular Plates using Energy Principle (Ritz method). Isotropic thin Rectangular plate having different twelve boundary conditions were analyzed and these boundary conditions were formed by combination of three major supports – Clamp, C; Simply supported, S; and Free, F.; General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von karman’s non-linear governing differential compatibility and equilibrium equations. Polynomial function instead of trigonometry function as was with previous researchers was used on the decoupled Von Karman’s equations to obtain particular stress and displacement functions respectively. Non-linear total potential Energy was formulated using Von Karman equilibrium equation and Ritz method was deployed in this formulation. This equation was fully converted to potential energy by multiplying all the terms in it with displacement, w and the formed total potential energy, π consists of potential energy of internal forces and potential energy of external forces. This formulated total potential energy π, could give an accurate approximation of displacement field if the parameters were properly chosen. However, we assumed deflection, w to be ∆H1, and stress function, ɸ to be ∆2H2 and substituted into the formulated potential energy. H1 and H2 are profiles of the deflection and stress function respectively, and ∆ is deflection coefficient factor of the plate. Potential energy formulated contains deflection coefficient factor to the power of four. This potential energy was minimized by differentiating it partially with respect to coefficient factor reducing to cubic form.

A fast simulation method for hydraulic-fracture-network generation in fractured rock based on fully coupled XFEM

In considering the rapid decrease of stress away from the crack tip, a fast simulation method based on fully coupled XFEM is proposed to handle the low efficiency and poor convergence in simulating the generation of complex fracture networks. The weak forms of governing equations are discretized by the XFEM approximation of displacement field and FEM approximation of one-dimensional pressure field. The Newton-Raphson method is then adopted to solve the resulting fully coupled u-p discretization equations. After the validation of the numerical model by the propagation of a single fracture and interaction between hydraulic fracture and natural fracture, several numerical cases are adopted to illustrate the superiority of the proposed model in computational efficiency. The influence of several key factors on the propagation of hydraulic fracture in a formation containing regularly and randomly distributed natural fractures is investigated. Numerical results show that the orientation has the greatest influence on the pattern of the hydraulic-fracture network in formation with regularly distributed natural fractures. Higher natural fracture density, lower differential stress, and interface strength can contribute to the higher injection pressure, the longer total length of shear fracture, and the more complex fracture network.

Free-breathing motion-informed locally low-rank quantitative 3D myocardial perfusion imaging.

PurposeTo propose respiratory motion‐informed locally low‐rank reconstruction (MI‐LLR) for robust free‐breathing single‐bolus quantitative 3D myocardial perfusion CMR imaging. Simulation and in‐vivo results are compared to locally low‐rank (LLR) and compressed sensing reconstructions (CS) for reference.MethodsData were acquired using a 3D Cartesian pseudo‐spiral in‐out k‐t undersampling scheme (R = 10) and reconstructed using MI‐LLR, which encompasses two stages. In the first stage, approximate displacement fields are derived from an initial LLR reconstruction to feed a motion‐compensated reference system to a second reconstruction stage, which reduces the rank of the inverse problem. For comparison, data were also reconstructed with LLR and frame‐by‐frame CS using wavelets as sparsifying transform (ℓ1‐wavelet).Reconstruction accuracy relative to ground truth was assessed using synthetic data for realistic ranges of breathing motion, heart rates, and SNRs. In‐vivo experiments were conducted in healthy subjects at rest and during adenosine stress. Myocardial blood flow (MBF) maps were derived using a Fermi model.ResultsImproved uniformity of MBF maps with reduced local variations was achieved with MI‐LLR. For rest and stress, intra‐volunteer variation of absolute and relative MBF was lower in MI‐LLR (±0.17 mL/g/min [26%] and ±1.07 mL/g/min [33%]) versus LLR (±0.19 mL/g/min [28%] and ±1.22 mL/g/min [36%]) and versus ℓ1‐wavelet (±1.17 mL/g/min [113%] and ±6.87 mL/g/min [115%]). At rest, intra‐subject MBF variation was reduced significantly with MI‐LLR.ConclusionThe combination of pseudo‐spiral Cartesian undersampling and dual‐stage MI‐LLR reconstruction improves free‐breathing quantitative 3D myocardial perfusion CMR imaging under rest and stress condition.

Spatially curved functionally graded Timoshenko microbeams: A numerical study using IGA

In this paper, we present numerical investigations on static bending and free vibration behaviors of spatially curved functionally graded (FG) microbeam using isogeometric analysis (IGA). The material properties change continuously along the thickness direction. The small scale effects are captured using the modified couple stress theory while microbeam kinematics are described by Timoshenko beam theory, and the spatially curved microbeam is represented by the Frenet–Serret bases. The geometries of curved microbeams are exactly represented with the non-uniform rational B-spline (NURBS) basis functions, and the same basis functions are applied to obtain displacement field approximation. Numerical examples are presented and computed results are then compared with reference methods to show the accuracy of the developed method. The effects of some factors including material gradient factor, material length scale parameter, and boundary conditions on the mechanical behaviors of spatially curved FG Timoshenko microbeams are investigated.

A displacement correlation method for stress intensity factor extraction from 3D fractures in anisotropic materials

A numerical method to extract stress intensity factors (SIFs) from 3D fractures in anisotropic materials is presented. The formulation of a displacement correlation method that is valid for linear elastic materials with arbitrary symmetry in their elastic properties is given in detail. An algorithm for the numerical implementation of the method is also given. The methodology can take an approximate displacement field from any numerical method as input to the extraction process. However, to return accurate stress intensity factors an accurate numerical displacement field is required. In this work, this is provided by a Generalized Finite Element Method. Several verification problems show that the anisotropic displacement correlation method is robust and can extract accurate SIFs. The methodology is used in a parametric study of crack–crack interactions in a high-strength aluminum alloy to demonstrate a practical application. More than 200 simulations are run using the Generalized Finite Element Method, which are largely able to be scripted and ran without user intervention after one simulation is set up.

Zigzag Theory for Piezoelectric-Layer-Integrated Functionally Graded Material Plates

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Thermodynamically consistent thermoelastic plate and shell formulations for small deformation and small strain using non-classical continuum mechanics incorporating internal rotations

The work presented in a recent paper by the authors [35] for a thermodynamically consistent and kinematic assumption free plate and shell formulation for small deformation and small strain based on the conservation and balance laws of classical continuum mechanics (CCM) is extended here for non-classical continuum mechanics (NCCM). This formulation incorporates additional physics due to internal rotations that arise due to the deformation gradient tensor. This physics is neglected in CCM, hence is absent in the plate and shell formulation of reference [35]. Consideration of this new physics requires modifications of the current balance laws as well as consideration of a new balance law “balance of moment of moments” (BMM) [2, 3]. Cauchy stress tensor becomes non-symmetric. Cauchy moment tensor is conjugate to the symmetric part of the rotation gradient tensor which exists now due to new physics. Balance of angular momenta yields additional three differential equations as part of the mathematical model. The new balance law (BMM) establishes symmetry of the Cauchy moment tensor. The new physics considered here exists in all deforming solid continua as it is due to the deformation gradient tensor, but is ignored in CCM. The consequence of this new physics is additional stiffness, hence additional strain energy storage and change in the time history of displacements and stress field compared to formulations based on CCM. The basic mathematical model for the plate and shell deformation consists of conservation and balance laws in $$\mathbb {R}^3$$ based on NCCM incorporating internal rotations. The associated finite element formulations for obtaining the solution of the mathematical model consists of : (i) geometry of the plate or shell described by the flat or curved middle surface (as done conventionally) and nodal vectors locating the top and bottom faces of the plate/shell (ii) the displacement field approximation that is p-version hierarchical in the plane as well as in the transverse direction (iii) integral form is constructed using Galerkin Method with Weak Form (GM/WF) and the corresponding element equations. The formulation presented here remains valid and accurate for thin as well as thick plate/shell and naturally reduces to the formulation of reference [35] based on classical continuum mechanics. Model problem studies and comparisons with the studies based on CCM formulation [35] will be presented in a follow up paper.

Non-conforming interface coupling and symmetric iterative solution in isogeometric FE–BE​ analysis

In this paper, a three-dimensional (3D) isogeometric finite element (FE) and boundary element (BE) coupling approach based on non-uniform rational B-splines (NURBS) is developed to simulate the deformed body. Based on the geometric exactness and higher smoothness of isogeometric analysis (IGA), NURBS discretization is applied to the approximation of geometric representation, displacement field and traction field simultaneously. An improved power series expansion method (M-PSEM) is used to evaluate the singular integral and is suitable for high order (≥3) NURBS basis functions, which is the limitation of the traditional method. Adaptive integration scheme based on quadtree subdivision technique is adopted to enable the non-singular integration computation more flexible and efficient at optimal computational cost. The relationship between the tractions and nodal external forces on the interface is established by using the virtual work principle. On the other hand, displacement coupling constraints among the non-conforming (or incomplete coincident) surface meshes are constructed by a virtual knot insertion technique. The main advantages of this method are simplicity and robustness, as it is problem-independent and only depends on the NURBS meshes on both sides of the interface. Introducing an enhanced symmetric matrix related to the BE subdomain, the symmetric iterative coupling method (SICM) is extended to solve the final system coupling equation. The final coupling matrix obtained by this iterative method is symmetric, and its dimension is the same as that of the stiffness matrix of IGA finite element method (IGAFEM). Numerical examples are investigated to assess the accuracy and efficiency of the presented method.

Free vibration analysis of FG-CNTRC conical shell panels using the kernel particle Ritz element-free method

The free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell is carried out by using element-free kp-Ritz method. Based on the first-order shear deformation shell theory, the approximate displacement field is expressed by the shape function of the nuclear particle, and the governing equation is established. The material properties of conical shell plates are determined by rule of mixtures and change with the thickness. In this work, the convergence is studied in terms of the number of nodes, and the accuracy of the proposed method is verified by comparing the current solution with the solution in the literature from different aspects. The effects of volume fraction, boundary conditions, distribution patterns of CNTs, half vertex angle and radius thickness ratio on their free vibration frequency characteristics are discussed in detail.

Analysis and active control of geometrically nonlinear responses of smart FG porous plates with graphene nanoplatelets reinforcement based on Bézier extraction of NURBS

In this paper, we propose an effective computational approach to analyze and active control of geometrically nonlinear responses of functionally graded (FG) porous plates with graphene nanoplatelets (GPLs) reinforcement integrated with piezoelectric layers. The key concept behind this work is to utilize isogeometric analysis (IGA) based on Bézier extraction technique and C0-type higher-order shear deformation theory (C0-HSDT). By applying Bézier extraction, the original Non-Uniform Rational B-Spline (NURBS) control meshes can be transformed into Bézier elements which allow us to inherit the standard numerical procedure like the standard finite element method (FEM). In this scenario, the approximation of mechanical displacement field is calculated via C0-HSDT whilst the electric potential field is considered as a linear function across the thickness of each piezoelectric sublayer. The FG plate includes internal pores and GPLs dispersed into metal matrix either uniformly or non-uniformly along plate’s thickness. To control responses of structures, the top and bottom surfaces of FG plate are firmly bonded with piezoelectric layers which are considered as sensor and actuator layers. The geometrically nonlinear equations are solved by Newton-Raphson iterative procedure and Newmark’s integration. The influence of porosity coefficient, weight fraction of GPLs as well as external electrical voltage on the geometrically nonlinear behaviors of plate structures with various distributions of porosity and GPLs are thoroughly investigated. A constant displacement and velocity feedback control approaches are then adopted to actively control geometrically nonlinear static and dynamic responses, where structural damping effect is taken into account, based on a closed-loop control with sensor and actuator layers.

Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

We propose a new finite-volume based approach to the solution of Saint Venant’s torsion problems of homogeneous and composite prismatic bars subjected to pure torsion. The semi-inverse approach is employed to construct an approximate displacement field characteristic of pure torsion within the subvolumes of the beam’s discretized cross section such that the governing differential equation for the warping function is locally satisfied in each subvolume in an integral sense. Orthotropic subvolumes are admitted in the formulation to enable analysis of bars comprised of heterogeneous microstructures, including composite materials with orthotropic shear moduli. Continuity of both displacements and tractions across subvolumes’ interfaces is satisfied in a surface-average sense, together with traction-free lateral boundary conditions. Closed-form expressions for the stiffness matrix elements that relate surface-averaged displacements to tractions are provided for ease of programming. The convergence and accuracy of the proposed method are assessed and verified upon comparison with the exact elasticity solutions for prismatic homogeneous and composite bars of rectangular cross section and the corresponding finite-difference solution for the homogeneous bar. The utility of the developed solution methodology is first illustrated by analyzing torsion of homogeneous T and box beams, focusing on the singular behavior at the re-entrant corners and the concomitant membrane analogy’s applicability. Then, composite cross sections reinforced by angle braces as well as continuous fiber-reinforced wraps with graded shear moduli are investigated with the aim of mitigating the re-entrant corner shear stress singularities, not easily analyzed using the finite-element or finite-difference approaches.

Hypersingular boundary integral equations for plane orthotropic elasticity in terms of first-order stress functions

This paper is intended to present an implementation of the hypersingular boundary integral equations in terms of first-order stress functions for stress computations in plane orthotropic elasticity. In general, the traditional computational technique of the boundary element method used for computing the stress distribution on the boundary and close to it is not as accurate as it should be. In contrast, the accuracy of stress computations on the boundary is greatly increased by applying the hypersingular integral equations. Contrary to the method in which the solution is based on an approximation of displacement field, here the first-order stress functions and the rigid body rotation are the fundamental variables. An advantage of this approach is that the stress components can be obtained directly from the stress functions, there is, therefore, no need for Hooke's law, which should be used when they are computed from displacements. In addition, the computational work can be reduced when the stress distribution is computed at an arbitrary point on the boundary. The numerical examples presented prove the efficiency of this technique.

First-order shear deformation analysis of rectilinear orthotropic composite circular plates undergoing transversal loads

This work outlines the elastic bending analysis of transversely loaded shear-deformable rectilinear orthotropic composite annular plates. The load condition discussed in the paper, along with the displacement constraints, are derived by the composite bolted joints theoretical reference model – this analytical solution is a necessary effort for the obtainment of a custom finite element capable of simulating this kind of joints with high accuracy and limited computational effort. Firstly, the constitutive equations of the family of plate under investigation are obtained in the framework of the First-order Shear Deformation Plate Theory. The described methodology is founded on the application of the virtual displacements principle and its solution is performed according to Ritz method after the writing of displacement field approximation functions fulfilling the boundary conditions. The three unknown displacement components are obtained for different case studies concerning rectilinear orthotropic composite annular plates featuring various slenderness ratios, shape factors and stacking sequences. The outcomes comparison with FE numerical solutions evidences a high degree of fidelity. The presented results demonstrate that this enhanced version of the Ritz analytical solution method, which accounts for the composite plate shear deformability, can be more effectively employed to describe the displacement field of composite plates connected by a bolted joint in the area surrounding the bolt.

A novel formulation for static and buckling analysis of plates using coupled element free Galerkin-finite strip (EFG-FS)

• Coupling element free Galerkin and finite strip (EFG-FS) method is proposed for the static and buckling analysis of thin plates. • The buckling analysis of a skew plate under shear loading is presented using EFG-FS method. • The buckling analysis of a plate with a hole of complicated shape at the center is presented using EFG-FS method. • The analysis times are decreased by using the EFG-FS method than the EFG method . The element free Galerkin (EFG) method has been successfully used for a wide variety of problems over the last few decades, but due to its high cost of computing, it is preferred to combine it with other methods having low computational costs such as the Finite Strip (FS) method. So, in this paper, a novel method is proposed by coupling EFG and classical FS methods for the static and buckling analysis of thin plates. The present method can have advantages of both EFG and classical FS methods. It is very easy to implement, and essential boundary conditions are easily enforced in FS sub-domain. The EFG method is based on the moving least-squares (MLS) approach. Therefore, the shape function does not have the Kronecker delta function property. In order to satisfy the combined conditions of displacement compatibility, a modified variational form is used based on the Lagrange multiplier method. In the FS sub-domain, three angular functions are used to approximate displacement field in the longitudinal direction of each strip. Finally, numerical examples are presented with different boundary conditions to demonstrate the effectiveness of the present coupling method.

Vibroseismic protection of heavy mining machines, buildings and structures

In the article, the authors analyze main issues of protecting buildings and structures against vibroseismic effects with the help of system of elastomeric blocks, which are characterized by high vertical stiffness, low shear stiffness, high energy dissipation and ability to center vertical load; besides, they feature high reliability with no risk of sudden failure. Results of static and dynamic tests of parametric series of elastomeric blocks used for protection of residential buildings against vibrations are presented. Design of pile with anti-vibration rubber supports is described. Calculation of elastomeric blocks under static compression is presented. Components of the stress-strain state of the thin-layer elements were investigated, and rate of the thin rubber layer compression under the action of vertically applied force was calculated. In numerical calculation, weak compressibility of rubber was simulated by moment force of the finite element for weakly compressible materials, which assumes triple approximation of displacement fields, deformation components, and volume change function. The numerical solution was obtained by the finite element method for different radius-thickness ratios in case of geometrically nonlinear elastic and viscoelastic deformation.

Dynamic Gradients, Mobile Layers, Tg Shifts, Role of Vitrification Criterion, and Inhomogeneous Decoupling in Free-Standing Polymer Films

The force-level Elastically Collective Nonlinear Langevin Equation theory of activated relaxation in glass-forming free-standing thin films is re-visited to improve its treatment of collective elasticity effects. The naive cut off of the isotropic bulk displacement field approximation is improved to explicitly include spatial anisotropy with a modified boundary condition consistent with a step function liquid-vapor interface. The consequences of this improvement on dynamical predictions are quantitative but of significant magnitude and in the direction of further speeding up dynamics and further suppressing Tg. The theory is applied to thin films and also thick films to address new questions for three different polymers of different dynamic fragility. Variation of the vitrification time scale criterion over many orders of magnitude is found to have a minor effect on changes of the film-averaged Tg relative to its bulk value. The mobile layer length scale grows strongly with cooling and correlates in a nearly linear manner with the effective barrier deduced from the corresponding bulk isotropic liquid alpha relaxation time. The theory predicts a new type of spatially inhomogeneous "dynamic decoupling" corresponding to an effective factorization of the total barrier into its bulk temperature-dependent value multiplied by a function that only depends on location in the film. The effective decoupling exponent grows as the vapor surface is approached. Larger reductions of the absolute value of Tg shifts in thin polymer films are predicted for longer time vitrification criteria and more fragile polymers. Quantitative no-fit-parameter comparisons with experiment and simulation for film-thickness-dependent Tg shifts of PS and PC are in reasonable accord with the theory, including a nearly 100 K suppression of Tg in 4 nm PC films. Predictions are made for polyisobutylene thin films.

B-spline finite element approach for the analysis of thin-walled beam structures based on 1D refined theories using Carrera unified formulation

In the current study, 1D refined beam theories on the basis of Carrera Unified Formulation (CUF) are combined with Isogeometric approach (IGA) for the static and free vibration analysis of thin-walled beam structures. The B-spline basis functions utilized in IGA are employed to approximate displacement field due to their interesting attributes in analysis. N-Order Taylor-like expansions are utilized in the framework of CUF which presents finite element matrices in the form of fundamental nuclei that is independent of the type and order of expansion. Higher-order B-spline basis functions attenuate the effect of shear locking properly and higher-order theories presented by CUF are free from Poisson locking phenomenon and utilizations of shear correction factor. Several numerical results including both solid and thin-walled structures are investigated and it is shown that coupling IGA and CUF ends in a suitable methodology to analyze beam structures.

Free Vibration Analysis of Fibre-Metal Laminated Beams via Hierarchical One-Dimensional Models

This paper presents a free vibration analysis of beams made of fibre-metal laminated beans. Due to its attractive properties, this class of composites has gained more and more importance in the aeronautic field. Several higher-order displacements-based theories as well as classical models (Euler-Bernoulli’s and Timoshenko’s ones) are derived, assuming Carrera’s Unified Formulation by a priori approximating the displacement field over the cross section in a compact form. The governing differential equations and the boundary conditions are derived in a general form that corresponds to a generic term in the displacement field approximation. The resulting fundamental term, named “nucleus”, does not depend upon the approximation order N, which is a free parameter of the formulation. A Navier-type, closed form solution is used. Simply supported beams are, therefore, investigated. Slender and short beams are considered. Three- and five-layer beams are studied. Bending, shear, torsional, and axial modes and frequencies are presented. Results are assessed for three-dimensional FEM solutions obtained by a commercial finite element code using three-dimensional elements showing that the proposed approach is accurate yet computationally effective.

Static analysis of functionally graded nanocomposite sandwich plates reinforced by defected CNT

In this paper, stress distribution and deflection in sandwich plates with functionally graded nanocomposite face sheets have been carried out by a first order shear deformation theory (FSDT) based mesh-free method. The sandwich plates are assumed resting on Winkler-Pasternak elastic foundation and the nanocomposite is reinforced by three types of defected carbon nanotubes (CNTs) which are compared with pristine CNT-reinforced nanocomposite. A multiscale modeling with molecular dynamics (MD) simulations in the nanoscale and micromechanical approach of Halpin-Tsai are used to calculate the elastic constants of the nanocomposite. In the mesh-free analysis, moving least squares (MLSs) shape functions are used to approximate displacement field and the transformation method is used for imposition of essential boundary conditions. The effects of Stone-Wales (SW) and vacancy defect configurations of CNTs, number of defects, CNT distribution and volume fraction, boundary conditions and geometric dimensions are investigated on the static analysis of the sandwich plates. It is observed that when CNT volume fraction is 5%, six vacancy defects in CNT configuration can increase the deflection of the sandwich plate up to 41.67%.