Element Displacement Field Research Articles

High resolution digital image correlation using proper generalized decomposition: PGD‐DIC

SUMMARYA new method is proposed to measure the finite element (FE) displacement field from a deformed image in comparison with a reference one. In opposition to standard FE approaches, the unknown displacement is sought as a sum of products of separated dimension functions. With the problems in each dimension being uncoupled, the method involves only one‐dimensional meshes and one‐dimensional problems. An algorithm that builds successive best rank‐one approximations is proposed and integrated into the nonlinear iterations of the correlation problem. Although the method can be applied to spaces of any dimension, this paper focuses on two‐dimensional images. Many synthetic examples are provided to evaluate the performance of the method. In addition, it is shown that, even with this separated representation, the introduction of a regularization operator is convenient. The latter makes it possible to perform a pixel‐wise measure with huge computational savings. Copyright © 2012 John Wiley & Sons, Ltd.

Crack problems in a viscoelastic medium using enriched finite element method

Crack problems in 2-D and 3-D linear viscoelastic media are numerically investigated using the enriched finite element method (enriched FEM). In the same spirit as for the enriched elements, the viscoelastic crack-tip asymptotic displacement field is obtained by using the viscoelastic–elastic correspondence principle and is transformed into the polynomial displacement field of regular elements. Thereby, the time-dependent unknowns in the asymptotic displacement field related to the crack strain energy release rate are introduced as the additional degree of freedoms (DOFs) of enriched elements. With these treatments, the incremental enriched FEM formulations in time domain are derived according to the Boltzmann superposition principle. The present method shows that the time-dependent strain energy release rate can be easily obtained from the extra DOFs. Several illustrative examples are given and good agreement between the present results and those from analytical and other numerical methods is confirmed. The method developed in this paper is capable of solving crack problems in viscoelastic media with complicated geometry, loading and constraint conditions.

Assessment of Measuring Errors in Strain Fields Obtained via DIC on Planar Sheet Metal Specimens with a Non-Perpendicular Camera Alignment

The determination of strain fields based on displacement components obtained via 2D-DIC is subject to several errors that originate from various sources. In this contribution, we study the impact of a non-perpendicular camera alignment to a planar sheet metal specimen’s surface subject to biaxial loading conditions. The errors are estimated in a numerical experiment. To this purpose, deformed images - that were obtained by imposing finite element (FE) displacement fields on an undeformed image - are numerically rotated for various Euler angles. It is shown that a 3D-DIC stereo configuration induces a substantial compensation for the introduced image-plane displacement gradients. However, higher strain accuracy and precision are obtained - up to the level of a perfect perpendicular alignment - in a proposed ”rectified” 2D-DIC setup. This compensating technique gains benefit from both 2D-DIC (single camera view, basic amount of correlation runs, no cross-camera matching nor triangulation) and 3D-DIC (oblique angle compensation).

Orthogonal basic deformation mode method for zero-energy mode suppression of hybrid stress elements

A set of basic deformation modes for hybrid stress finite elements are directly derived from the element displacement field. Subsequently, by employing the so-called united orthogonal conditions, a new orthogonalization method is proposed. The resulting orthogonal basic deformation modes exhibit simple and clear physical meanings. In addition, they do not involve any material parameters, and thus can be efficiently used to examine the element performance and serve as a unified tool to assess different hybrid elements. Thereafter, a convenient approach for the identification of spurious zero-energy modes is presented using the positive definiteness property of a flexibility matrix. Moreover, based on the orthogonality relationship between the given initial stress modes and the orthogonal basic deformation modes, an alternative method of assumed stress modes to formulate a hybrid element free of spurious modes is discussed. It is found that the orthogonality of the basic deformation modes is the sufficient and necessary condition for the suppression of spurious zero-energy modes. Numerical examples of 2D 4-node quadrilateral elements and 3D 8-node hexahedral elements are illustrated in detail to demonstrate the efficiency of the proposed orthogonal basic deformation mode method.

Finite Element Analysis of Von-Mises Stress Distribution in a Spherical Shell of Liquified Natural Gas (Lng) Pressure Vessels

This research work investigated the modeling of Von Mises stress in LNG Spherical Carbon Steel Storage tank using assumed displacement Finite Element analysis based on shallow shell triangular elements. Using equations of elasticity, constant thickness carbon steel spherical storage tanks were subjected to different loading conditions. This paper stresses the need for proper definition of shallow element using sector angles to obtain the shallowness. The shallow spherical triangular element has five degrees of freedom at each of its corner node, which are the essential external degrees of freedom. The assumed displacement fields of these shallow triangular elements satisfied the exact requirement of rigid body modes of motion. The FORTRAN 90 programming language was used for the programme coding to solve finite element equations resulting from the model while Von Mises stresses distribution within the spherical storage tank shell subjected to different internal pressures were determined. The results showed that the use of non-shallow elements due to improper sector angles resulted in unreliable results while real shallow elements produced results that tallied with ASME Section VIII Div 1, Part UG values.

Error estimation in measuring strain fields with DIC on planar sheet metal specimens with a non-perpendicular camera alignment

The determination of strain fields based on displacement components obtained via 2D-DIC is subject to several errors that originate from various sources. In this contribution, we study the impact of a non-perpendicular camera alignment to a planar sheet metal specimen's surface. The errors are estimated in a numerical experiment. To this purpose, deformed images – that were obtained by imposing finite element (FE) displacement fields on an undeformed image – are numerically rotated for various Euler angles. It is shown that a 3D-DIC stereo configuration induces a substantial compensation for the introduced image-plane displacement gradients. However, higher strain accuracy and precision are obtained – up to the level of a perfect perpendicular alignment – in a proposed “rectified” 2D-DIC setup. This compensating technique gains benefit from both 2D-DIC (single camera view, basic amount of correlation runs, no cross-camera matching nor triangulation) and 3D-DIC (oblique angle compensation). Our conclusions are validated in a real experiment on SS304.

Modeling of large deformation – Large sliding contact via the penalty X-FEM technique

Abstract In this paper, an extended finite element method is employed to simulate the presence of discontinuities caused by the contact surface. In X-FEM, the need for mesh adaption to interface is neglected and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the elements. The modified level set technique and the Heaviside enrichment function are employed to approximate the discontinuous displacement field of elements located on the contact surface. The penalty method is used to impose the contact constraints and establish the non-penetration condition. An efficient numerical algorithm is employed to model the large deformation contact behavior based on the node-to-segment technique. Finally, numerical examples are presented to demonstrate the accuracy and capability of proposed X-FEM technique in modeling of large deformation – large sliding contact problems.

A study of multiscale wavelet-based elements for adaptive finite element analysis

This paper is concerned with the construction of multiscale wavelet-based elements using lifting scheme. In deriving the computational formulation of multiscale elements of B-spline wavelet on the interval (BSWI), the element displacement field represented by the coefficients of wavelets expansion in wavelet space is transformed into the physical degree of freedoms (DOFs) in finite element space via the corresponding transformation matrix. Then 2D C0 type multiscale BSWI elements are derived to fulfill the nesting approximation of wavelet finite element method (WFEM). The wavelet-based adaptive algorithm shares the approaches involved in adaptive classical finite element methods. Numerical results indicate that the present multiscale wavelet-based elements are suit for adaptive finite element analysis, especially for singularity problems in engineering. The convergence shown in numerical examples demonstrates the reliability of the elements.

Study of systematic errors in strain fields obtained via DIC using heterogeneous deformation generated by plastic FEA

In this article, systematic errors that arise in the derivation of strain fields based on displacement fields obtained by digital image correlation (DIC) are analysed. Attention is paid to errors that arise from different implementations of the DIC technique. In particular, we investigate the influence of the shape function, the interpolation order and the subset size on the derived strains. In addition, we focus on errors that can be directly attributed to the derivation of the strain fields, e.g. the strain-window size and the strain-window interpolation order. The errors are estimated using numerically deformed images that were obtained by imposing finite element (FE) displacement fields on an undeformed image yielding plastic deformation of the specimen. This FE procedure simulates realistic experimental heterogeneous deformations at various load steps. It is shown that the errors on the strain fields can be substantially reduced if conscious choices in the abovementioned implementations are made.

Assessment of measuring errors in DIC using deformation fields generated by plastic FEA

In this article, systematic errors that arise from different implementations of digital image correlation (DIC) techniques are analyzed. In particular, we investigate the influence of the adopted correlation function, the interpolation order, the shape function and the subset size on the derived displacements. These errors are estimated using numerically deformed images that were obtained by imposing finite element (FE) displacement fields on an undeformed image yielding plastic deformation of the specimen. This FE procedure simulates realistic experimental heterogeneous deformations at various load steps. It is shown that DIC is able to reproduce these displacements up to a satisfactory level if conscious choices in the above-mentioned implementations are made.

Nonlinear analysis of composite beams with partial shear interaction by means of the direct stiffness method

This paper presents a modelling technique for the nonlinear analysis of composite steel-concrete beams with partial shear interaction. It extends the applicability of two stiffness elements previously derived by the authors using the direct stiffness method, i.e. the 6DOF and the 8DOF elements, to account for material nonlinearities. The freedoms are the vertical displacement, the rotation and the slip at both ends for the 6DOF stiffness element, as well as the axial displacement at the level of the reference axis for the 8DOF stiffness element. The solution iterative scheme is based on the secant method, with the convergence criteria relying on the ratios of the Euclidean norms of both forces and displacements. The advantage of the approach is that the displacement and force fields of the stiffness elements are extremely rich as they correspond to those required by the analytical solution of the elastic partial interaction problem, thereby producing a robust numerical technique. Experimental results available in the literature are used to validate the finite element proposed in the paper. For this purpose, those reported by Chapman and Balakrishnan (1964), Fabbrocino et al. (1998, 1999) and Ansourian (1981) are utilised; these consist of six simply supported beams with a point load applied at mid-span inducing positive bending moment in the beams, three simply supported beams with a point load applied at mid-span inducing negative bending moment in the beams, and six two-span continuous composite beams respectively. Based on these comparisons, a preferred degree of discretisation suitable for the proposed modelling technique expressed as a function of the ratio between the element length and depth is proposed, as is the number of Gauss stations needed. This allows for accurate prediction of the nonlinear response of composite beams.

Numerical prediction of crack propagation and crack widths in concrete structures

The paper deals with the numerical prediction of crack propagation and crack widths in concrete structures. The early stages of cracking, characterized by distributed micro-cracking, are described by a smeared crack model with rotating crack directions, whereas the subsequent formation of a macro-crack is described by embedding a discontinuity in the displacement field of the respective finite elements. This so-called crack model with delayed embedded discontinuities allows correcting initial mispredictions of the crack direction during crack evolution. It accounts for crack opening in the direction normal to the crack and for relative tangential displacements of the crack faces with transfer of shear forces across the crack faces. After validation of the model by a well-known test series on combined normal and shear loading of concrete specimens, it is applied to two problems of practical relevance in engineering practice. The first one deals with the load-carrying behavior of steel anchors embedded into concrete and the second with the structural behavior of pre-damaged concrete slabs strengthened by different types of concrete overlays.

Digital holographic cameras and data processing for remote monitoring and measurements of mechanical parts

AbstractFast growing technology and requirements for testing of different types of materials and devices require new methods and systems for investigation of their parameters. Among the quantities of high interest are shape, deformation, roughness, local materials constants, displacement, and strain fields of elements under load. In the paper, we present novel solutions for digital holographic cameras, which allow for remote monitoring and measurement of the above mentioned quantities at small mechanical objects or at restricted areas of interest at big structures. The systems have compact design, “black box” measurement approach, and allows for fast and accurate measurements performed directly at the element and often in outdoor environment. The principles of digital and optoelectronic reconstruction and phase manipulation are described together with the exemplary measurement results obtained by means of the cameras presented.

A new wavelet-based thin plate element using B-spline wavelet on the interval

By interacting and synchronizing wavelet theory in mathematics and variational principle in finite element method, a class of wavelet-based plate element is constructed. In the construction of wavelet-based plate element, the element displacement field represented by the coefficients of wavelet expansions in wavelet space is transformed into the physical degree of freedoms in finite element space via the corresponding two-dimensional C1 type transformation matrix. Then, based on the associated generalized function of potential energy of thin plate bending and vibration problems, the scaling functions of B-spline wavelet on the interval (BSWI) at different scale are employed directly to form the multi-scale finite element approximation basis so as to construct BSWI plate element via variational principle. BSWI plate element combines the accuracy of B-spline functions approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples are studied to demonstrate the performances of the present element.

Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.

Vibration analysis of beams with non‐local foundations using the finite element method

AbstractIn this paper, a non‐local viscoelastic foundation model is proposed and used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local foundation models the reaction of the non‐local model is obtained as a weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two‐node beam elements. However, for non‐local elasticity or damping, nodes remote from the element do have an effect on the energy expressions, and hence the damping and stiffness matrices. The expressions of these direct and cross‐matrices for stiffness and damping may be obtained explicitly for some common spatial kernel functions. Alternatively numerical integration may be applied to obtain solutions. Numerical results for eigenvalues and associated eigenmodes of Euler–Bernoulli beams are presented and compared (where possible) with results in literature using exact solutions and Galerkin approximations. The examples demonstrate that the finite element technique is efficient for the dynamic analysis of beams with non‐local viscoelastic foundations. Copyright © 2007 John Wiley & Sons, Ltd.

The construction of 1D wavelet finite elements for structural analysis

Adopting the scaling functions of B-spline wavelet on the interval (BSWI) as trial functions, a new finite element method (FEM) of BSWI is presented. Instead of traditional polynomial interpolation, scaling functions at the certain scale have been adopted to form the shape functions and construct wavelet-based elements. Unlike the process of wavelets added directly in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The transformation matrix is the key to construct wavelet-based elements freely as long as we can ensure its non-singularity. Then, classes of C0 and C1 type elements are constructed. And the lifting scheme of BSWI elements is also discussed. The numerical examples indicate that the BSWI elements have higher efficiency and precision than traditional finite element method in solving 1D structural problems especially for geometric nonlinear, variable cross-section and loading cases.

An advanced FE analysis of PSC shell structures incorporating tendon-induced deformation-dependent loads

In the analysis of prestressed concrete (PSC) structures, the nonconservativeness of the prestressing effect has been often neglected or treated in a simple manner. However, the tendon-induced deformation-dependent loads can disturb the fast convergence to the correct solution and sometimes make the solution diverge without elegant formulations, when the PSC structures undergo large deformations. To solve this problem, an advanced numerical analysis procedure is proposed for the PSC shell structures. Geometrically nonlinear finite element analysis is performed for the tendon-induced equivalent loads that are considered here as the follower loads. In many cases, prestressing tendons are arbitrarily arranged inside a concrete structure with some eccentricities with respect to the mid-surface. In this study, therefore, combined effects of the improved displacement field of the shell elements with the eccentrically applied follower loads are investigated and consistently derived with the form of the load correction stiffness matrix (LCSM). Numerical examples show that the proposed procedures can correctly predict the geometrically nonlinear response of the PSC shell structures up to large deformations with the apparent contribution of the derived LCSM and exhibit much faster convergence behavior.

Response Sensitivity for Nonlinear Beam–Column Elements

Response sensitivity is needed for simulation applications such as optimization, reliability, and system identification. The exact response sensitivity of material nonlinear beam-column elements is derived for the displacement- and force-based formulations. For displacement-based beam.column elements the response sensitivity is straightforward to compute because the displacement field is specified along the element. A new approach is presented for computing the response sensitivity of force-based beam-column elements, in which the displacement field is not specified. In this approach, the response sensitivity depends on the derivative of unbalanced section forces because the element displacement field changes with the element state. Example nonlinear static analyses of steel and reinforced concrete structural systems verify the exact response sensitivity for force-based elements using the new approach.

The quadratic finite element/strip with generalized degrees of freedom and their application

This paper presents a quadratic finite element with generalized degrees of freedom (GDOF) and a quadratic finite strip with GDOF based on the principle that the local displacement fields of elements should be compatible with the global displacement field of the corresponding systems. Firstly, a global displacement field is developed using quadratic B-spline functions. Secondly, local displacement fields of elements and strips are constructed, employing multi-term interpolation polynomials of second degree. Making the local displacement fields of elements and strips be compatible with their corresponding global displacement fields, respectively, models of the finite element with GDOF and the finite strip with GDOF are accordingly generated. On the other hand, the quadratic B-spline function is convenient to derive the explicit form of characteristic equations for computing model because of its lower order, and accordingly simplifies the computation, compared with cubic and quintic spline functions. Several numerical examples demonstrate the accuracy, simplicity and versatility of the present element and strip in the analysis of thin-walled structures.