Equations Of Linear Elasticity Research Articles

Response of a plate in turbulent channel flow: Analysis of fluid–solid coupling

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid–structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier–Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. The resonant vibration is stronger at the third natural frequency than at the first natural frequency. We explain this behavior by comparing the fluid and solid length scales. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectrum of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the sources reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be ≈0.75δ. The integrated contribution of the overlap and outer regions together to the plate response is comparable to that from the buffer region for Reτ=180 and exceeds the buffer region contribution for Reτ=400. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the intensity of the sources peaks in the buffer region and the wall-normal width of the sources extend well into the outer region of the channel.

Analysis of crack parameters under pure Mode II loading by modified exponential matrix method

A new analytical method for calculating the Stress Intensity Factor (SIF) under Mode II loading is presented in the framework of two-dimensional stationary elasticity. Starting from the linear elasticity equations, the crack problem is formulated as a system of Hamiltonian equations, solved based on a new method for determining eigensystems, called Exponential Matrix Method (EMM). The obtained eigensolutions satisfy the adjoint symplectic orthogonality by means of performing an angular stress variation with respect to the radial stress. The linear combination of these eigensolutions then allows a good description of the expected solution. Related to one of the asymptotic expansion coefficients, the SIF under Mode II Loading is calculated using the trapezoidal integration method (TIM). The implemented methodology validation is carried out by comparing the obtained results with those provided by the literature review.

Three-dimensional topology optimization of thermal-fluid-structural problems for cooling system design

In the present study, a topology optimization method of thermal-fluid-structural problems is researched to design the three-dimensional heat sink with load-carrying capability. The optimization is formulated as a mean temperature minimization problem controlled by Navier-Stokes (N-S) equations as well as energy balance and linear elasticity equations. In order to prevent an unrealistic and low load-carrying design, the power dissipation of the fluid device and the normal displacement on the load-carrying surface are taken as constraints. A parallel solver of multi-physics topology optimization problems is built-in Open Field Operation And Manipulation (OpenFOAM) software. The continuous adjoint method is adopted for the sensitivity analysis to make the best use of built-in solvers. With the developed tool, the three-dimensional (3D) thermal-fluid topology optimization is studied. It is found that the Darcy number, which is suitable for fluid design, may cause severe problems in thermal-fluid optimization. The structural features of 3D thermal-fluid-structural problems are also investigated. The “2D extruded designs” are helpful to improve the structural stiffness, and channels with a larger aspect ratio in high-temperature areas improve the cooling performance.

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

ABSTRACTWe propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.

Fluid structure interaction analysis of the operating Savonius wind turbine

In the last few decades vertical axis wind turbines (VAWTs) have gained increased interest among researchers worldwide. This paper aims to present full two-way Fluid Structure Interaction (FSI) approach to model operation of the Savonius-type VAWT. The developed methodology couples the set of Unsteady Reynolds Averaged Navier-Stokes Equations (URANS) with the unsteady linear elasticity equations and dynamic mesh approach to comprehensively describe the deformations of the turbine rotor. The obtained results provide detailed information on the unsteady flow fields around the operating wind turbine as well as deformations of a turbine rotor caused by an interaction between solid and flowing air. This allowed us to determine the influence of the unsteady deformations of a turbine rotor on the instantaneous and average performance of a wind turbine. Computations were carried out for the wind speed equal to 15 m/s and the tip speed ratio equal to 0.8. The simulations were performed using the commercial ANSYS software. The obtained results showed that deformation of the turbine rotor increases the wind turbine power coefficient by around 5%. Moreover, it was shown that the centrifugal force has a significant effect on the rotor deformation and turbine operation, hence it cannot be neglected.

Diffuse domain method for needle insertion simulations.

We present a new strategy for needle insertion simulations without the necessity of meshing. A diffuse domain approach on a regular grid is applied to overcome the need for an explicit representation of organ boundaries. A phase field function captures the transition of tissue parameters and boundary conditions are imposed implicitly. Uncertainties of a volume segmentation are translated in the width of the phase field, an approach that is novel and overcomes the problem of defining an accurate segmentation boundary. We perform a convergence analysis of the diffuse elastic equation for decreasing phase field width, compare our results to deformation fields received from conforming mesh simulations and analyze the diffuse linear elastic equation for different widths of material interfaces. Then, the approach is applied to computed tomography data of a patient with liver tumors. A three-class U-Net is used to automatically generate tissue probability maps serving as phase field functions for the transition of elastic parameters between different tissues. The needle tissue interaction forces are approximated by the absolute gradient of a phase field function, which eliminates the need for explicit boundary parameterization and collision detection at the needle-tissue interface. The results show that the deformation field of the diffuse domain approach is comparable to the deformation of a conforming mesh simulation. Uncertainties of tissue boundaries are included in the model and the simulation can be directly performed on the automatically generated voxel-based probability maps. Thus, it is possible to perform easily implementable patient-specific elastomechanical simulations directly on voxel data.

Subtraction singularity technique applied to the regularization of singular and hypersingular integrals in high-order curved boundary elements in plane anisotropic elasticity

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

Closed-form orthotropic constitutive model for aligned square array mesostructure

This work models the square array mesostructure found in fused filament fabrication parts using an orthotropic constitutive model and derives closed-form expressions for all nine effective elastic constants. The periodic void shape is modeled using a four point hypotrochoid curve with a single shape parameter that controls the sharpness of the points. In-plane elastic constants (Eyy, Ezz, vyz, Gyz) are derived by solving the 2D elasticity equations using the complex variable method of elasticity. The antiplane shear constants (Gxy and Gxz) are derived from a complex variable formulation of the antiplane shear equations. The remaining elastic constants (Exx, vxy, and vxz) are derived by directly solving the linearelasticity equations. We compare our results to unit cell finite element simulations. The simulations match the closed-form expressions exactly for Exx, vxy, and vxz. For the remaining elastic constants, the difference between closed-form model and simulation increases as porosity increases. Tensile testing specimens at various porosity values under 12% are fabricated to test six of the nine elastic constants. Experiments on Exx, vxy, and vxz agree well with closed-form predictions, differing by less than 2.6% for the tested specimen types. Experiments on Eyy, Ezz, and Gxy generally agree well, with larger differences at higher porosity values. The difference is less than 6% for all tested specimen types except for the Ezz specimen with a porosity of 0.113, which had a difference of 21%. Several reasons for the discrepancy are postulated, including non-ideal void shape and interfacial bond stiffness.

Modeling of photoelastic imaging of mechanical stresses in transparent solids mimicking kidney stones.

Theoretical and numerical models were developed to calculate the polariscopic integrated light intensity that forms a projection of the dynamic stress within an axisymmetric elastic object. Although the model is general, this paper addressed its application to measurements of stresses in model kidney stones from a burst wave lithotripter for stone fragmentation. The stress was calculated using linear elastic equations, and the light propagation was modeled in the instantaneous case by integrating over the volume of the stone. The numerical model was written in finite differences. The resulting images agreed well with measured images. The measured images corresponded to the maximum shear stress distribution, although other stresses were also plotted. Comparison of the modeled and observed polariscope images enabled refinement of the photoelastic constant by minimizing the error between the calculated and measured fields. These results enable quantification of the stress within the polariscope images, determination of material properties, and the modes and mechanisms of stress production within a kidney stone. Such a model may help in interpreting elastic waves in structures, such as stones, toward improving lithotripsy procedures.

Stress constrained thermo-elastic topology optimization based on stabilizing control schemes

This article proposes two effective stabilizing control schemes for addressing the stress constrained thermo-elastic topology optimization in a non-uniform temperature field. Based on the density interpolation scheme, two linear elastic equations for coupling a thermo-elastic problem are considered. For comparison, different topology problem formulations for minimizing compliance or volume subject to stress constraints are solved. By virtue of a stabilization transform method, two stabilizing control schemes combined with the grouped aggregation method are developed to handle the challenging difficulties stemming from the local nature of highly nonlinear stress constraints. Moreover, the adjoint method is adopted to perform the sensitivity analysis. The design variables are updated by utilizing the method of moving asymptotes. The results of several typical numerical examples verify the validity of the proposed methodology, including the present stabilizing control schemes which can be employed to obtain clear topological design and fast convergence rate for thermo-elastic coupling problems. Meanwhile, compliance minimization design with stress constraints is appropriate to achieve balance between stress level and stiffness.

Hyperelastic finite deformation analysis with the unsymmetric finite element method containing homogeneous solutions of linear elasticity

SummaryA recent unsymmetric 4‐node, 8‐DOF plane finite element US‐ATFQ4 is generalized to hyperelastic finite deformation analysis. Since the trial functions of US‐ATFQ4 contain the homogenous closed analytical solutions of governing equations for linear elasticity, the key of the proposed strategy is how to deal with these linear analytical trial functions (ATFs) during the hyperelastic finite deformation analysis. Assuming that the ATFs can properly work in each increment, an algorithm for updating the deformation gradient interpolated by ATFs is designed. Furthermore, the update of the corresponding ATFs referred to current configuration is discussed with regard to the hyperelastic material model, and a specified model, neo‐Hookean model, is employed to verify the present formulation of US‐ATFQ4 for hyperelastic finite deformation analysis. Various examples show that the present formulation not only remain the high accuracy and mesh distortion tolerance in the geometrically nonlinear problems, but also possess excellent performance in the compressible or quasi‐incompressible hyperelastic finite deformation problems where the strain is large.

Linear superposition method (LSM) for solving stress tensor fields and displacement vector fields: Application to multiple pressure-loaded circular holes in an elastic plate with far-field stress

This study presents a novel, linear superposition method (LSM) to compute the stress tensor field and displacement vector field in a homogeneous elastic medium with an unlimited (but finite) number of circular cylindrical holes. The displacement field and the associated stress concentrations are due to a far-field stress. The method allows for the hole-centers to occur in arbitrary locations, and the hole-radii may vary over a wide range (but holes may not overlap). The holes may also induce additional elastic displacement due to internal pressure loading that will affect the local stress field, which is fully accounted for in the method. Each hole may be loaded by either equal or individual pressure loads. The underlying algorithms and solution methodology are explained and examples are given for a variety of cases. Selected case study examples show excellent matches with results obtained via independent methods (photo-elastics, complex analysis, and discrete volume solution methods). The LSM provides several advantages over alternative methods: (1) Being closed-form solutions, infinite resolution is preserved throughout, (2) Being grid-less, no time is lost on gridding, and (3) fast computation times. The specific examples of LSM applications to the multi-hole problem developed here, allow for an unlimited number of holes, with either equal or varying radii, in arbitrary constellations. The solutions further account for variable combinations of far-field stress and pressure loads on individual holes. The method can be applied for either plane strain or plane stress boundary conditions. A constitutive equation for linear elasticity controls the stress field solutions, which can be scaled for the full range of Poisson's ratios and Young moduli possible in linear elastic materials.

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline- (WEB-spline-) based discrete solution for the 2D Navier-Lamé equation of linear elasticity with a different type of mixed boundary condition called CA,B boundary condition. Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

The integration of structural mechanics into microstructure solidification modelling

In situ structural mechanics are an often neglected area when modelling alloy microstructure during solidification, despite the existence of practical examples and studies which seem to indicate that the interaction between thermal or mechanical stresses and microstructure can have a significant impact on its evolution and hence the final properties at a macroscopic level. A bespoke structural mechanics solver using the finite volume method has been developed to solve the linear elasticity equations, with design choices being made to facilitate the coupling of this solver to run in situ with an existing solidification model. The accuracy of the structural mechanics solver is verified against an analytic solution and initial results from a fully coupled system are presented which demonstrate in a fundamental example that the interaction between structural mechanics and a solidifying dendrite can lead to a significant change in growth behaviour.

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.

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

SummaryThis article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right‐hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user‐defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time‐harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high‐dimensional parameter space.

A high-order discontinuous Galerkin method for simulating incompressible fluid-thermal-structural interaction problems

This paper presents a framework for the application of the discontinuous Galerkin(DG) finite element method to the multi-physics simulation of the solid thermal deformation interacting with incompressible flow problems in two-dimensions. Recent applications of the DG method are primarily for thermoelastic problems in a solid domain or fluid-structure interaction problems without heat transfer. Based on a recently published conjugate heat transfer solver, the incompressible Navier-Stokes equation, the fluid advection-diffusion equation, the Boussinesq term, the solid heat equation and the solid linear elastic equation are solved using an explicit DG formulation. A Dirichlet-Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux computed at interface quadrature points in the fluid-solid interface. Formal hp convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved for each solver. The algorithm is then further validated against several existing benchmark cases including the in-plane loaded square, the Timoshenko Beam, the laminated beam subject to thermal-loads and the lid-driven cavity with a flexible bottom wall. The computational effort demonstrates that for all cases the highest order accurate algorithm has several magnitudes lower error than the second-order schemes for a given computational effort. It is a strong justification for the development of such high order discretisations. The solver can be employed to predict thermal deformation of a structure due to convective and conductive heat transfer at low Mach, such as chip deformation on a printed circuit board, wave-guide structure optimization, thermoelectric cooler simulation, and optics mounting method verification.

SILICA and its process zone

AbstractThis article reviews recent works concerning the damage area size ahead of a crack front in oxide glasses. Literature exemplifies the importance of understanding the crack propagation dynamics in oxide glasses with a number of papers concerning dynamic and stress corrosion cracking in complex (and simplified) oxide glasses. In general, the linear elastic fracture mechanic equation breaks down near the crack front, in a zone frequently called the process zone. The question is how large is this zone near the crack tip? This paper reviews four different techniques to capture the process zone size in pure silica: Atomic force microscopy imaging, neutron diffraction, photon capture, and molecular dynamics simulations. Velocities of the crack front range between stress corrosion cracking dynamics and dynamic fracture, thus providing a better understanding of how the process zone depends on the velocity of the crack front.

Analysis of the Flow Dynamics of Blood Through Viscoelastic Constricted Artery

In this paper, we focused on the flow through an axisymmetric constricted artery of the pulmonary region to study the condition of stenosis. Theory of dust particles suspended in gas is applied on blood flow through the artery, where the “particles” represent “cells” suspended in plasma. The flow is governed by two dimensional Navier–Stokes’ equations by including Darcy–Forchheimer drag force caused by non-Darcian effect. The material of the artery is approximated as a linear elastic and simplest rheological equation that includes viscosity and elasticity (considered lung as a Voigt body) is used. Effect of various parameters, such as Reynolds number (Re), Forchheimer number ($$F_s$$), Darcy number (Da), aspect ratio ($$\beta $$), shape factor ($$S_f$$), porosity ($$\epsilon $$), aerodynamic diameter ($$d_p$$), bulk compression ($$\phi $$) of elasticity, shear ($$\eta $$) and bulk ($$\zeta $$) coefficients of parenchymal viscosity are obtained on the radial and axial velocities of blood and particles graphically. We found that the fluid (blood) and particle (cells suspended in plasma) velocities along both the axes (axial and radial) increase by increasing Reynolds number, the pulsating amplitude, aspect ratio, and porosity of walls. While by increasing Forchheimer number, velocities of blood and particles in both the axes decreases gradually. The present analysis is also indicate that the viscoelasticity of walls are affected by the amplitude of pulsatile flow of blood and for a large value of amplitude, the viscoelastic effect decreases.

A one-way coupled model for the vibration of tuning fork-based trace gas sensors driven by a thermoacoustic wave

We present a single computational model for both quartz-enhanced photoacoustic spectroscopy and resonant optothermoacoustic detection trace gas sensors. These sensors employ a quartz tuning fork to detect the acoustic pressure and thermal waves generated when a laser excites a trace gas. The model is based on a coupled system of equations developed by Morse and Ingard for pressure and temperature in a fluid. The pressure and temperature solutions drive the resonant vibration of the tuning fork, which is modeled using the equations of linear elasticity. At high ambient pressure, excellent agreement is obtained with laboratory experiments. This result provides the first quantitative match between a fully computational simulation and experiments for a QEPAS sensor. Such a model could ultimately facilitate sensor design optimization. At low ambient pressure (less than 60 Torr), quantitative agreement is obtained after reweighting the contributions from the pressure and thermal components of the signal. While this result is a substantial improvement over previous results in which a scaling factor was required to obtain agreement at any ambient pressure, at low pressures, it appears that a more accurate physical model may be required to match experimental data.