Interior Displacement Research Articles

Nodal Accuracy Improvement Technique for Linear Elements with Application to Adaptivity

In the finite element method, the conventional linear elements have long been precluded, due to their low accuracy of nodal displacements, from the analysis of super-convergence and adaptivity via the element energy projection (EEP) technique. To overcome this problem, in this paper, a nodal accuracy improvement technique is proposed for linear elements in 1D to 3D problems. In this method, a residual nodal load vector is derived with the conventional EEP solution, and a simple back-substitution process can generate the improved nodal displacements without changing the global stiffness matrix. Subsequently, an improved EEP scheme for linear elements is proposed based on the improved nodal displacements. Finally, by using the improved EEP solution as an error estimator, a two-phased adaptive algorithm is presented. Numerical examples show that the accuracy of nodal displacements is improved from the second-order convergence to the fourth-order convergence by using the nodal accuracy improvement technique, and the EEP solutions for element interior displacements are improved from the second-order convergence to the third-order convergence by using the improved EEP scheme. Therefore, the improved EEP scheme can be effectively used as an error estimator in adaptivity analysis for linear elements, which turns out to be efficient in general and even outperforms cubic elements for singularity problems.

Effect of simulated earthquake loading on radon exhalation from uranium tailings dam.

Uranium tailings sand will continuously release radon-222. When the external condition changes, the exhalation of radon will also change. Thus, radon is being recommended as a tracer for dam damage assessment. When an earthquake is simulated on the uranium tailings dam with a shaking table test and the change in radon concentration is measured, it is observed that the earthquake causes micro-fissures in the uranium tailings dam, which aggregate to form fractures. During the process, the radon concentration will climb dramatically, as will the radon exhalation rate. To verify that the radon monitoring date is accurate, the acceleration response, surface displacement, and interior displacement are all monitored. The results show that radon can be utilized as a tracer to evaluate uranium tailings dam damage.

An oblique circular cylinder element for 3D interfacial cracks in composites

This study further extends a developed symplectic analytical singular element (SASE) library to include three-dimensional (3D) cracks in dissimilar materials. We construct a trial symplectic eigen solution for 3D bimaterial interfacial cracks and use it as the crack tip enrichment to define the interior displacement field of a new singular element. The symplectic approach is applied for the numerical modelling of 3D bimaterial cracks for the first time. In the derivation, a non-standard shape of the 3D crack-tip element, i.e., a cylinder with two oblique ends with arbitrary orientations, is considered to better fit the mesh of curved 3D cracks. It is proven that the proposed 3D element has improved the accuracy and is still convenient to use. The present contribution has provided a new insight for choosing effective and efficient crack tip enrichments of 3D interfacial cracks, and is also an important step forward of the SASE library.

Visualization of water channeling and displacement diversion by polymer gel treatment in 3D printed heterogeneous porous media

Polymer gels are widely applied to increase the sweep efficiency in mature oil fields facing severe water channeling problems after years of water flooding. The formation of water channeling and successful polymer gel treatment are rooted in complex pore-scale displacement and diversion. Thus, accurate characterization of the pore space and direct visualization of the interior displacement processes in real reservoir rocks such as water flooding, polymer gel injection, and chase floods are crucial in determining the controlling mechanisms of water channeling and polymer gel diversion. In this study, 3D X-ray micro-computed tomography (μCT) was used to characterize the porous structures of three distinct levels in water-flooded reservoir sandstones. A heterogeneous digital model was designed, which comprised representative layers of the obtained porous structures. Then, 3D-printing technology was used to manufacture a transparent model based on the digital prototype. A series of displacement experiments were conducted in the printed model to directly visualize the flow behaviors in the actual pore geometries. In the experiments, the model was first saturated with white oil and flooded by formation water until the outlet discharged only the water phase. Then, an immediately prepared polymer gel was injected. After three days of gelation, an antidilution polymer was injected to sweep the residual oil in the model. Finally, a surfactant flooding agent was injected designed to further increase the oil recovery. The entire four-step displacement process was directly visualized, and the saturation of each injection and the oil recovery were calculated. During water flooding, preferential flooding channels formed in the highest water-flooded porous layers due to the low viscosity ratio between the injected formation water and the displaced white oil, as well as the permeability contrast in the heterogeneous model. The injected polymer gel mainly followed the preferential flooding channels because of the low resistance force in these channels. After gelation, the polymer gel blocked the main preferential channels and diverted the chase floods to nearby pores. A desirable diversion of the polymer gel highly depends on its propagation path during injection and effective blocking after gelation. Experimental visualization and displacement mechanism analysis indicate the efficacy of the flow diversion by the polymer gel treatment in the heterogeneous porous model. The results provide useful information on the displacement behavior in heterogeneous porous structures, and the conclusions apply to similarly scaled systems at the reservoir scale.

A stacked frequency approach for inhomogeneous time-dependent MRE: an inverse problem for the elastic shear modulus

Abstract We derive and analyse a new way to calculate the shear modulus of an inhomogeneous elastic material from time-dependent magnetic resonance elastography (MRE) measurements of its interior displacement. Even with such a rich data source, this is a challenging inverse problem because the coefficient of the shear modulus in the governing equations can be small (or potentially zero). Our approach overcomes this by combining different data sets into an overdetermined matrix–vector equation. It uses finite differences to approximate space derivatives and a Fourier interpolant in time, and we do not need to assume that the inhomogeneous material is ‘locally homogeneous’. Crucially, our construction ensures that the computed value of the (real) shear modulus is real: approximation methods based on the frequency domain version of the problem often give a complex shear modulus for the elastic case and this can be hard to interpret, especially if its imaginary part dominates. We carry out careful numerical tests on a one (space) dimensional analogue of the problem and on experimental MRE data for an inhomogeneous gel phantom.

Study on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problems

The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind of problem by using the degenerate kernels in terms of the elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can’t be clearly examined. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.

Fundamental-solution-based hybrid finite element with singularity control for two-dimensional mixed-mode crack problems

In this work, a fundamental-solution-based hybrid finite element method is presented for modeling mixed-mode cracks in two-dimensional (2D) isotropic elastic media. In the method, a double-variable hybrid functional for each element is formulated to derive element stiffness equation in terms of nodal displacements, which includes line integrals along the element boundary only. The element interior displacement field is approximated using a linear combination of displacement fundamental solutions at different source locations, while the independent element frame displacement field is approximated by one-dimensional shape function interpolation. To correctly model the behavior of crack-tip displacement, the discontinuous quarter-point crack-tip singular hybrid element formulation is developed, so that crack tip stress intensity factors can be easily evaluated by the near-tip displacement method. Three numerical examples of internal cracks in 2D elastic domains are presented to show the efficiency of the proposed method.

New hybrid-Trefftz Mindlin–Reissner plate elements for efficiently modeling the edge zones near free/SS1 edges

PurposeThe purpose of this paper is to propose a new finite element method (FEM) solving strategy for efficient analysis of the challenging edge effect problem in plate structures. Its main ideas are to develop special-purpose plate element models to effectively simulate the behaviors in the plate’s edge zones near free/SS1 edges.Design/methodology/approachThese new elements are developed based on the hybrid-Trefftz element method. During their construction procedures, the analytical solutions of the edge effect problem, which are in exponential forms, are used to enhance the interior displacement fields. Besides, the Lagrangian multipliers are introduced into the modified hybrid-Trefftz functional for considering the stress resultant constraints at free/SS1 edges. Thus, these elements theoretically possess the abilities to correctly capture the very steep gradients of the resultant distributions in the boundary layers.FindingsThese new specialized hybrid-Trefftz plate elements can very efficiently solve the edge effect problem with high accuracy, even when distorted meshes are used. Moreover, because these elements’ construction procedures contain only boundary integrals, the computation expense for accurately integrating the exponential trial functions can be considerably saved.Originality/valueThis work presents an alternative novel idea for using the FEM to more effectively handle the local stress problems by incorporating the use of the analytical trial functions.

Green’s-function-based-finite element analysis of fully plane anisotropic elastic bodies

In the paper, an anisotropic Green’s function based hybrid finite element was developed for solving fully plane anisotropic elastic materials. In the present hybrid element, the interior displacement and stress fields were approximated by the linear combination of anisotropic Green’s functions derived by Lekhnitskii formulation, the element frame fields were constructed by the interpolation of general shape functions widely used in the conventional finite element, and then they are linked by a new double-variable hybrid functional. Because the approximated interior fields exactly satisfied the governing equations related to anisotropic elasticity, all integrals in the present hybrid functional were performed along the element boundary and theoretically arbitrary hybrid polygonal element can be constructed. Finally, the present hybrid polygonal element with four edges was verified by making comparison of numerical results and exact solutions in a cantilever composite beam made with angled lamina.

Embalmed in the Camera's Glowing Formaldehyde: On John Yau's Hollywood Asians

Abstract The poetry of John Yau has examined identity and exile through complex portraits of pop culture icons. His portraits of Peter Lorre as Mr. Moto draw upon postmodern techniques from the visual arts, particularly the aesthetic strategies of Jasper Johns, and his poetic experiments embody the thematics of interior exile and displacement in dense surrealist texts. His dominant strategy is to undermine representation in ways similar to both philosophies of deconstruction and postmodern aesthetics, but without losing sight of the cultural politics of ethnicity.

3D Deformation and Strain Analysis in Uniaxial Compressive Coal Using Digital Volumetric Speckle Photography

Digital Volumetric Speckle Photography (DVSP), which is the extension of 2D digital speckle photography, is applied to measure the interior deformation and strain of the object by 2-step 3D FFT. For using FFT, DVSP has some advantages in computational efficiency. With the special loading setup for Micro-CT, the coal specimen under different uniaxial loadings is scanned, and the volumetric images are acquired. By using DVSP, the interior 3D displacements of the specimen are obtained, and then the equivalent strains and volumetric strains are estimated. Based on the interior displacement and strain mappings, the process of deformation or strain localization in the coal specimen is studied.

Special Elements for Composites Containing Hexagonal and Circular Fibers

In this paper, unidirectional fiber reinforced composites with periodic square array of circular and hexagonal fibers is studied by a novel fundamental-solution-based hybrid finite element model. Due to the periodicity of composites, a representative unit cell containing a single fiber with circular or hexagonal cross section is taken into consideration and analyzed using the proposed hybrid finite element model. In the present numerical model, special polygonal fiber elements with arbitrary number of sides are developed by coupling the independent element interior and frame displacement fields. The element interior displacement fields are approximated by the combination of fundamental solutions to prior satisfy the governing equation of the problem, so that the domain integral appeared in the weak-form hybrid functional in terms of dual variables is converted into boundary integrals. Independently the element frame displacement fields are approximated by the conventional shape functions to guarantee the continuity of adjacent elements. Following this, special polygonal fiber elements are constructed to reduce mesh effort in the fiber region and achieve good accuracy with fewer elements. Finally, numerical tests are carried out for assessing the performance of the present special elements.

Stability in the linearized problem of quantitative elastography

The goal of quantitative elastography is to identify biomechanical parameters from interior displacement data, which are provided by other modalities, such as ultrasound or magnetic resonance imaging. In this paper, we analyze the stability of several linearized problems in quantitative elastography. Our method is based on the theory of redundant systems of linear partial differential equations. We analyze the ellipticity properties of the corresponding PDE systems augmented with the interior displacement data; we explicitly characterize the kernel of the forward operators and show injectivity for particular linearizations. Stability criteria can then be deduced. While joint reconstruction of all parameters suffers from non-ellipticity even for more measurements, our results show stability of the separate reconstruction of shear modulus, pressure and density; they indicate that singular strain fields should be avoided, and show how additional measurements can help in ensuring stability of particular linearized problems.

Numerical implementation of local effects due to two-dimensional discontinuous loads using special elements based on boundary integrals

Abstract In this paper, special-purpose elements are developed for solving local effects caused by discontinuous loads such as concentrated forces, line loads and patch loads applied in plane elastic structures. During the derivation of the special-purpose elements, the interior displacement and stress fields are composed of two parts: (1) the homogeneous solution part, which is represented by a linear combination of fundamental solutions at a number of source points outside the element domain; and (2) the particular load-dependent part, which is analytically represented by suitable local solutions. Meanwhile the independent frame displacements defined over the element boundary are approximated by conventional shape functions. The linkage between the two independent fields is established through use of a newly constructed hybrid variational functional, in which discontinuous loads are treated as generalized body forces. Using the property of delta function, the domain integral associated with discontinuous loads in the variational functional can be removed. The advantage of such special-purpose elements is that a large element, independent of the location of discontinuous loads, can be used to avoid the requirement of mesh refinement in the vicinity of the area with local loads. Numerical experiments are carried out to verify the special-purpose elements and to investigate their effectiveness in terms of mesh reduction and accuracy.

A new special element for stress concentration analysis of a plate with elliptical holes

Special hole elements are presented for analyzing the stress behavior of an isotropic elastic solid containing an elliptical hole. The special hole elements are constructed using the special fundamental solutions for an infinite domain containing a single elliptical hole, which are derived based on complex conformal mapping and Cauchy integrals. During the construction of the special elements, the interior displacement and stress fields are assumed to be the combination of fundamental solutions at a number of source points, and the frame displacement field defined over the element boundary is independently approximated with conventional shape functions. The hybrid finite element model is formulated based on a hybrid functional that provides a link between the two assumed independent fields. Because the fundamental solutions used exactly satisfy both the traction-free boundary conditions of the elliptical hole under consideration and the governing equations of the problems of interest, all integrals can be converted into integrals along the element boundary and there is no need to model the elliptical hole boundary. Thus, the mesh effort near the elliptical hole is significantly reduced. Finally, the numerical model is verified through three examples, and the numerical results obtained for the prediction of stress concentration factors caused by elliptical holes are extremely accurate.

Improving arrival time identification in transient elastography

In this paper, we improve the first step in the arrival time algorithm used for shear wave speed recovery in transient elastography. In transient elastography, a shear wave is initiated at the boundary and the interior displacement of the propagating shear wave is imaged with an ultrasound ultra-fast imaging system. The first step in the arrival time algorithm finds the arrival times of the shear wave by cross correlating displacement time traces (the time history of the displacement at a single point) with a reference time trace located near the shear wave source. The second step finds the shear wave speed from the arrival times. In performing the first step, we observe that the wave pulse decorrelates as it travels through the medium, which leads to inaccurate estimates of the arrival times and ultimately to blurring and artifacts in the shear wave speed image. In particular, wave ‘spreading’ accounts for much of this decorrelation. Here we remove most of the decorrelation by allowing the reference wave pulse to spread during the cross correlation. This dramatically improves the images obtained from arrival time identification. We illustrate the improvement of this method on phantom and in vivo data obtained from the laboratory of Mathias Fink at ESPCI, Paris.

FEM Analysis of 1-3 Cement Based Piezoelectric Composites

The simulating software ANSYS is applied to carry out the coupling analysis of the 1-3 cement based piezoelectric composites. The interior displacement and strain distribution were obtained and discussed. The influences of material parameters such as thickness, relative dielectric constant, Poisson’s ratio and Yang’s modulus on the composites were analyzed based on the FEM model. The results indicate that active strain and displacement of the 1-3 cement based piezoelectric composites is mainly concentrated on the piezoelectric ceramic rods. The performance of the 1-3 cement based piezoelectric composites can be improved efficiently by increasing the thickness, Poisson's ratio, and reducing the Yang's modulus, the relative dielectric constant.

Shear modulus reconstruction in dynamic elastography: time harmonic case

This paper presents a direct inversion approach for reconstructing the elastic shear modulus in soft tissue from dynamic measurements of the interior displacement field during time harmonic excitation. The tissue is assumed to obey the equations of nearly incompressible, linear, isotropic elasto-dynamics in harmonic motion. A finite element discretization of the governing equations is used as a basis, and a procedure is outlined to eliminate the need for boundary conditions in the inverse problem. The hydrostatic stress (pressure) is also reconstructed in the process, and the effect of neglecting this term in the governing equations, which is common practice, is considered. The approach does not require iterations and can be performed on sub-regions of the domain resulting in a computationally efficient method. A sensitivity study is performed to investigate the detectability of abnormal regions of different size and shear modulus contrast from the background. The algorithm is tested on simulated data on a two-dimensional domain, where the data are generated on a very fine mesh to get a near exact solution, then downsampled to a coarser mesh that is similar to the spatial discretization of actual data, and noise is added. Results showing the effect of the hydrostatic stress term and noise are presented. A reconstruction using MR measured experimental data involving a tissue-mimicking phantom is also shown to demonstrate the algorithm.

A boundary element method for solving 2-D and 3-D static gradient elastic problems: Part I: Integral formulation

A boundary element formulation is developed for the static analysis of two- and three-dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects. The simple gradient elastic theory of Aifantis expressed in the framework of Mindlin’s general theory is used to model this material behaviour. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution for both two- and three-dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here. It is found that for a well-posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. Explicit expressions for interior displacements and stresses in integral form are also presented. All the kernels in the integral equations are explicitly provided.

Circumferential resonance modes of solid elastic cylinders excited by obliquely incident acoustic waves.

When an immersed solid elastic cylinder is insonified by an obliquely incident plane acoustic wave, some of the resonance modes of the cylinder are excited. These modes are directly related to the incidence angle of the insonifying wave. In this paper, the circumferential resonance modes of such immersed elastic cylinders are studied over a large range of incidence angles and frequencies and physical explanations are presented for singular features of the frequency-incidence angle plots. These features include the pairing of one axially guided mode with each transverse whispering gallery mode, the appearance of an anomalous pseudo-Rayleigh in the cylinder at incidence angles greater than the Rayleigh angle, and distortional effects of the longitudinal whispering gallery modes on the entire resonance spectrum of the cylinder. The physical explanations are derived from Resonance Scattering Theory (RST), which is employed to determine the interior displacement field of the cylinder and its dependence on insonification angle.