Hermite Element Research Articles

Hermite element for vibration analysis of thin-walled beams with higher-order beam theory

Hermite element for vibration analysis of thin-walled beams with higher-order beam theory

Simulating Thin Shells by Bicubic Hermite Elements

In this study, we present the bicubic Hermite element method (BHEM), a new computational framework devised for the elastodynamic simulation of thin-shell structures. The BHEM is constructed based on quadrilateral Hermite patches, which serve as a unified representation for shell geometry, simulation, collision avoidance, as well as rendering. Compared with the commonly utilized linear FEM, the BHEM offers higher-order solution spaces, enabling the capture of more intricate and smoother geometries while employing significantly fewer finite elements. In comparison to other high-order methods, the BHEM achieves conforming C1 continuity for Kirchhoff–Love (KL) shells with minimal complexity. Furthermore, by leveraging the subdivision and convex hull properties of Hermite patches, we develop an efficient algorithm for ray-patch intersections, facilitating collision handling in simulations and ray tracing in rendering. This eliminates the need for laborious remodeling of the pre-existing surface as the conventional approaches do. We substantiate our claims with comprehensive experiments, which demonstrate the high accuracy and versatility of the proposed method.

DIC-Net: Upgrade the performance of traditional DIC with Hermite dataset and convolution neural network

Digital image correlation (DIC) is a non-contact optical method that tracks the speckle pattern on specimen surface to calculate the displacement and strain by image correlation algorithm. Although the traditional DIC method can conveniently measure surface deformation, it still has many limitations: (1) the accuracy of displacement and strain calculation needs to be improved in the case of high deformation gradient; (2) under match or over-match can hardly be avoided when the filters are used to reconstruct smooth displacement or strain field, and (3) boundary effect remains unresolved in computing the deformation near the boundary of region of interest or the discontinuous area (e.g. area near crack tip or crack face). Recently, the deep learning based DIC (Deep-DIC) has revealed its attractive ability in handling above issues in traditional DIC, and impressive results have been achieved. The mean value of the absolute error (MAE) on the test set has been optimized to 0.0361 pixels using existing Deep-DIC approaches, which are accompanied by a real-time measurement speed. The network structure and training dataset are two key factors for the deep learning method. However, the current working networks have been modified from other image tasks and cannot fully adapt to the demands of the DIC tasks, and the dataset they generated still has evident flaws, limiting the method's accuracy and generalization performance which is utilized to assess performance on samples outside the training set. In this paper, we firstly proposed a new Hermite dataset that is created by using the high-order Hermite element to take account more complex deformation, then a new network architecture designed for the DIC task has been developed to extract richer deformation features. A test set of 3216 examples containing six different modes of displacement is used to compare the performance of our network with others. The proposed DIC-Net-d achieves the lowest MAE in the test set. Meanwhile, in the Star5 image sets from DIC-Challenge, the proposed DIC-Net-d achieves a spatial resolution of 17.25 pixels and a noise level of 0.0136 which outperforms existing traditional and non-traditional methods. Finally, the strain network trained by our Hermite dataset is also successful in predicting the strain field of Star6 in the DIC challenge. The experiment results show the superiority of the proposed Hermite dataset and new network with respect to other Deep-DIC methods.

Convergence analysis of the Galerkin finite element method for the fourth-order Rosenau equation

In this paper, the Galerkin finite element method (FEM) for solving the fourth-order Rosenau equation is proposed with the bicubic Hermite element. The existence and uniqueness of the approximate solution is demonstrated through the Browder fixed point theorem and the convergent result of order O(h2) in H2-norm is derived for the semi-discrete scheme. The linearized fully-discrete scheme is constructed and its error estimation of order O(h2+τ2) in H2-norm is deduced. Finally, some numerical results are provided to confirm our theoretical analysis. Here and later h and τ denote the mesh size and the time step, respectively.

A practical and effective regularized polynomial smoothing (RPS) method for high-gradient strain field measurement in digital image correlation

Digital image correlation (DIC) is an effective tool for studying the deformation of tested materials. However, high-gradient strain field (HSF) measurement remains a challenging issue in DIC, because multifold factors have a significant effect on the first correlation calculation step of the original displacement and the exact strain reconstruction in the second post-processing step. In this study, we focus on the second step. The formerly proposed local Hermite has been proven as an accurate and robust method, yet substantial programming work is involved in constructing a Hermite element. To solve this issue, a regularized polynomial smoothing (RPS) method is proposed, which replaces the Hermite function of the polynomial. The RPS applies the same kernel function as the point-wise least-squares (PLS) method for easy implementation. Tikhonov regularization and the generalized cross-validation (GCV) function are also used to decrease unavoidable noise and weaken the dependence on window size. The experiments demonstrate that, in the matched/overmatched case, Tikhonov regularization in RPS can greatly decrease the degree of morbidity caused by ill-posed problems in the PLS method, thereby ensuring smoothness and stability of the solution. Moreover, the influences of window size and polynomial order of both the RPS and PLS are also closely studied. In conclusion, the proposed RPS provides a practical, robust and accurate method for HSF measurement.

Measurement of vibrations of composite wings using high-order finite element beam

The purpose of this study is to show the influence of bending-torsion coupling on natural frequencies and mode shapes of aircraft wings by using two finite element beam formulations. The bending-torsion coupling parameters are the geometric parameter (distance between the mass axis and elastic axis of the cross-section of the beam) and the material coupling due to laminated composites. Cubic and high-order Hermite finite element interpolations are presented in this study, in order to show the influence of geometric and material coupling on natural frequencies and mode shapes. Starting by the governing partial differential equations of motion for the coupled bending-torsion beam with the bending and torsion equations, the Galerkin’s method is used with high-order finite element interpolation to obtain the high-order Hermitian shape functions. The mass and stiffness matrices are obtained using the kinetic energy and potential energy, respectively. The beam finite element has two nodes, the cubic element has three degrees of freedom at each end (transvers displacement, slope and torsion), where the high order element has five degrees of freedom at each end (transvers displacement, slope, curvature, gradient of curvature and torsion). The mass matrix contains geometric coupling terms and the stiffness matrix contains terms of material bending-torsion coupling. The obtained results using cubic and high-order finite element Euler-Bernoulli beam formulations are compared for a free vibration analysis of Goland metallic wing (geometric coupling) and validated with Dynamic Stiffness Method for composite wings.

Local Hermite (LH) Method: An accurate and robust smooth technique for high-gradient strain reconstruction in digital image correlation

High-gradient strain field (HSF) measurement is attracting more and more attention from engineering and scientific fields, where digital image correlation (DIC) is widely adopted to calculate the dramatically changing displacement fields on stressed samples. Accurate reconstruction of reliable strain field from noisy displacement field is one of the key issues, since strain calculation is highly sensitive to displacement noise. The formally proposed improved Hermite finite element smoothing method (IHFESM) and fast Hermite element method (FHEM) are effective global smoothing techniques based on Tikhonov regularization. However, they involve complex meshing and global regularization matrix assembly processes. In this work, we present a local version of FHEM, called local Hermite (LH) method, in which one Hermite element is taken as a smoothing window centered on one displacement data, and the whole field is handled point-wisely. Two types of LH method are obtained by employing Hermite element with two types of continuities, i.e. C1 and C2. The smoothed displacement and strain are consequently computed from the regularized local surface as done in FHEM. Two kinds of simulated noisy displacement fields and one high-gradient displacement field obtained from shape memory alloy (SMA) tensile test are used to compare the performance of the proposed LH methods with the widely used point-wise least-squares (PLS) method. Experiments show that LH method with C2 element is nearly insensitive to window size, which is a quite useful property for practical engineering test without much prior knowledge. More importantly, it generally produces lower error than PLS in resolving local HSF. Thus, LH is a robust and accurate method which is highly recommended for practical HSF measurement.

Fast Hermite element method for smoothing and differentiating noisy displacement field in digital image correlation

In our previous work, an improved Hermite finite element smoothing method (IHFESM) combined with the well-known Tikhonov regularization was proposed to smooth the noisy displacement field directly calculated by digital image correlation (DIC). Even though IHFESM could reconstruct reliable strain field for arbitrary region of interest (ROI), it still suffers from three defects in practice: (i) for the large scale problems, it involves high computational burden to find the optimum regularization parameter within the generalized cross-validation (GCV) function, since the inverse and the trace of two large matrices are frequently evaluated, (ii) the search range is too wide to find the desired optimum regularization parameter quickly, and (iii) for the arbitrary ROI, the complex two dimensional meshing routines are required to locate and mesh the invalid regions into triangular elements. In current work, to overcome above deficiencies, we propose the fast Hermite element method (FHEM). The proposed FHEM avoids the redundant computation in IHFESM by decomposing the resultant two small matrices to speed up the original matrix inverse and trace computation. To narrow down the search range of regularization parameter, the magnitude of the closeness term and roughness penalty term in error function are balanced by a new parameter. The calculation of global regularization matrix is also sped up by using simple formulation without any sophisticated meshing routines. Experiments show that the FHEM is at least 50 times as fast as IHFESM with similar accuracy, and it is recommended to be adopted to smooth and differentiate the noisy displacement field for DIC in practice.

Accuracy of Advection Calculation by Hermite Element with Complete Cubic Polynomials in Two Dimensional Problems

Accuracy of Advection Calculation by Hermite Element with Complete Cubic Polynomials in Two Dimensional Problems

Dynamic contact of a beam against rigid obstacles: Convergence of a velocity-based approximation and numerical results

Motivated by the study of vibrations due to looseness of joints, we consider the motion of a beam between rigid obstacles. Due to the non-penetrability condition, the dynamics is described by a hyperbolic fourth order variational inequality. We build a family of fully discretized approximations of this problem by combining some classical space discretizations with velocity based time-stepping algorithms for discrete mechanical systems subjected to unilateral constraints. We prove the stability and the convergence of these numerical methods. Finally we propose some examples of implementation using either Hermite or B-spline finite element approximations.

NURBS-enriched contact finite elements

A novel enrichment of finite elements for contact computations based on isogeometric analysis is presented. Each body is divided into two parts, an enriched contact surface and the bulk domain together with surfaces that are not in contact. The latter part comprises the large majority of the domain and is treated in the usual manner with standard linear basis function, preserving the efficiency of classical finite element techniques. The enriched contact surface is discretized using NURBS basis functions of at least second order, allowing for a locally differentiable surface representation. This avoids the problem of suddenly changing normal vectors between element boundaries on the contact surface. Following the concept of isogeometric analysis, the smooth basis functions are not only used to describe the surface geometry, but also to approximate the solution on the surface. This leads to higher accuracy in the contact integral evaluation.Numerical results are presented for 2D and 3D contact computations including frictionless sliding, adhesive peeling, and cohesive debonding. The presented contact element enrichment exhibits a major gain in numerical accuracy and stability without loss of efficiency compared to standard linear finite elements. The enrichment technique offers some advantages over Hermite and higher-order Lagrangian contact element enrichment techniques, such as locally differentiable surface representations in 3D, while featuring competitive accuracy and performance.

Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.

New approach to construct freeform surface by numerically differential formulation

We proposed a new design method for single freeform reflective (or refractive) surface tailored to redistribute the radiant flux onto a prescribed illumination pattern. Unlike the conventional optimization approaches based on the grid mapping, in this study we estimated each segmental freeform surface by locally solving a second-order differ- ential equation, which formulates the energy transportation between each domain cell. With finite element method via Hermite element, we validated a series of smooth reflective/refractive surfaces to reallocate the radiant flux from a point source toward a target plane with specific patterns. The proposed technique offers a large flexibility by varying the vectors of each ray with multiple refraction (or reflection), which imposes no restric- tion on the target distribution, collective solid angle, or even target topog- raphy. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attri- bution of the original publication, including its DOI. (DOI: 10.1117/1.OE.53.3.031307) Subject terms: freeform design; uniform illumination; nonimaging optics.

Analysis of Euler-Bernoulli Beam with Piecewise Quadratic Hermite Finite Elements

The piecewise quadratic Hermite polynomials are employed in the finite element context to analyze the static and free vibration behaviors of Euler-Bernoulli beam. The desirable C1 continuity is achieved for the piecewise quadratic Hermite element that is required for the numerical solution of the Galerkin weak form of Euler-Bernoulli beam. In contrast to the classical cubic Hermite element, the piecewise quadratic Hermite element has a piecewise constant curvature representation within each element and thus the integration of the stiffness matrix is trivial. Several benchmark problems are shown to demonstrate the convergence properties of the piecewise quadratic Hermite element. The frequency error of the beam free vibration with this quadratic Hermite element is derived as well. Numerical examples consistently verify the analytical convergence rates.

Analysis of Euler-Bernoulli Beam with Piecewise Quadratic Hermite Finite Elements

The piecewise quadratic Hermite polynomials are employed in the finite element context to analyze the static and free vibration behaviors of Euler-Bernoulli beam. The desirable C1 continuity is achieved for the piecewise quadratic Hermite element that is required for the numerical solution of the Galerkin weak form of Euler-Bernoulli beam. In contrast to the classical cubic Hermite element, the piecewise quadratic Hermite element has a piecewise constant curvature representation within each element and thus the integration of the stiffness matrix is trivial. Several benchmark problems are shown to demonstrate the convergence properties of the piecewise quadratic Hermite element. The frequency error of the beam free vibration with this quadratic Hermite element is derived as well. Numerical examples consistently verify the analytical convergence rates.

The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.

Improved Hermite finite element smoothing method for full-field strain measurement over arbitrary region of interest in digital image correlation

We investigate the finite element method and the Tikhonov regularization for smoothing noisy displacement field obtained from the digital image correlation (DIC). Among various finite elements, the C2 continuous Hermite finite element has been proven to be an effective high-order element which ensures both continuity and smoothness of the final strain field. However, the standard Hermite element is required to be rectangular, which cannot tackle the region of interest of arbitrary shape. To overcome this constraint, we propose the improved Hermite finite element smoothing method (IHFESM) to extend the generality of Hermite finite element that is able to measure the arbitrary region of interest. The IHFESM is implemented by eliminating the contribution to the global regularization matrix from the invalid region. The proposed method could be employed to smooth the noisy displacement field obtained by DIC, and reconstruct the reliable strain field by differentiation. The effectiveness of IHFESM is verified by two tension experiments of plates with three and four holes.

Upper Spectral Bound of Biharmonic Operator Eigenvalue Problem by Bicubic Hermite Element

This paper uses the bicubic Hermite element to compute the first four eigenvalues of the vibration problem of clamped plate by Matlab program and gives upper bound of the exact eigenvalues. Combing Matlab experiments on Morley element for lower spectral bound we can provide a range of the exact eigenvalues of biharmonic operator more accurately.

OS0716 XFEMのHermite型要素への適用(OS7-3 解析・評価・き裂II,OS-7 多軸応力下における疲労損傷および疲労き裂進展2)

The extended finite element method (XFEM) has an advantage that an arbitrary shaped crack and its extension can be treated easily. But it is also pointed out that the results of the fracture parameter have significant errors that come from the existence of 'blending elements'. In the standard formulation of XFEM, the nodes in the element including the crack tip have enriched additional degrees of freedom. The nodes in the outer region have usual degrees of freedom, i.e. 2 in the two dimensional problem. The elements in the region surrounding the crack site have nodes with enriched freedom at the crack side and normal freedom at the outer region side. These elements are called 'blending elements'. In this paper, a new Hermite element with three nodes has been formulated and applied to a crack problem with XFEM. Errors of fracture parameters calculated by the M-integral method were discussed from the point of view of the effect of the blending elements. Stress intensity factor of CCT specimen has a 10% of error with a simple formulation. It decreases down to 0.5% by considering the improvement of XFEM proposed by Shibanuma et. al.

SECOND-ORDER ANALYSIS FOR DESIGN OF GLASS-SUPPORTING AND PRE-TENSIONED TRUSSES

Glass wall supporting systems, comprising of pre-tensioned high strength trusses made from alloy steel bars, behave nonlinearly under wind and other loads. For a proper analysis and design, second-order and large deflection analyses are mandatory as the structural response like stress and deflections are highly dependent on the small movements and temperature effects. Also, careful consideration must be given to various effects such as support settlement due to live loads and material creep, temperature change, pre-tension force and wind pressure. It is not uncommon to see many glass wall systems fail in a simulated wind load test under the simulated design wind speed. This paper presents a rigorous analysis and design of this type of structural systems used in two projects in Hong Kong. The key message is the need to use "section strength check" in place of the member design adopted in the conventional effective length method for design of practical structures behaving nonlinearly since the prescriptive buckling check formulae become inaccurate or even invalid. The use of stability function allowing for initial imperfections in place of cubic Hermite element is proposed here because of the high degree of nonlinearity.