High-Accuracy 2D Digital Image Correlation Measurements with Bilateral Telecentric Lenses: Error Analysis and Experimental Verification

By comparing two digital images of a test planar specimen surface recorded in different configurations, two-dimensional digital image correlation (2D-DIC) provides full-field displacements to sub-pixel accuracy and full-field strains in the recorded images. For the 2D-DIC systems using an optical lens, a simple pinhole imaging model is commonly used to describe the linear relationship between the measured sensor plane displacements and the actual displacements in the object surface. However, in a practical measurement, various unavoidable disadvantageous factors, such as small out-of-plane motion of the test object surface occurred after loading, small out-of-plane motion of the sensor target due to the self-heating or temperature variation of a camera, and geometric distortion of the imaging lens, may seriously impair or slightly change the originally assumed linear correspondence. In certain cases, these disadvantages may lead to significant errors in displacements and strains measured by 2D-DIC. In this work, the measurement errors of 2D-DIC due to the above three disadvantageous factors are first described in detail. Then, to minimize the errors associated with these disadvantages, a high-accuracy 2D-DIC system using a bilateral telecentric lens is established. The performance of the established 2D-DIC system and other two 2D-DIC systems using a conventional lens and an object-side telecentric lens are investigated experimentally using easy-to-implement stationary, out-of-plane and in-plane rigid body translation tests. A detailed examination reveals that a high-quality bilateral telecentric lens is not only insensitive to out-of-plane motion of the test object and the self-heating of a camera, but also demonstrates negligible lens distortion. Uniaxial tensile tests of an aluminum specimen were also performed to quantitatively compare the axial and transversal strains measured by the proposed 2D-DIC system and those measured by strain gage rosettes. The perfect agreement between the two measurements further verifies the accuracy of the established 2D-DIC system.     

General framework for the identification of constitutive parameters from full-field measurements in linear elasticity

In this paper, several approaches available in the literature for identifying the constitutive parameters of linear elastic materials from full-field measurements are presented and their sensitivity to a white noise added to the data is compared. The first investigated approach is the virtual fields method (VFM). It is shown that the uncertainty of the parameters identified with the VFM when a white noise is added to the data depends on the choice of a relevant set of virtual fields. Optimal virtual fields exist, thus minimizing the uncertainty and providing the “maximum likelihood solution”. The other approaches investigated in this paper are based on finite element model updating (FEMU). It is proved that FEMU approaches actually yield equations similar to the ones derived from the VFM, but with nonoptimal sets of virtual fields. Therefore, the FEMU approaches do not provide the “maximum likelihood solution”. However, the uncertainty of FEMU approaches varies dramatically with the cost function to minimize. On one hand, the FEMU approach based on the “displacement gap” minimization yields equations which are very close to the ones of the VFM approach and therefore, its uncertainty is almost the same as the VFM one. On the other hand, it is shown that other approaches based on the “constitutive equation gap” minimization or the “equilibrium gap” minimization provide biased solutions. For all the approaches, very fast algorithms, converging in only two iterations, have been devised. They are finally applied to real experimental data obtained on an orthotropic composite material. Results confirm the success of two methods: the VFM approach which provides the “maximum likelihood solution” and the FEMU approach based on the “displacement gap” minimization.     

使用单彩色相机的单相机三维数字图像相关方法

介绍了一种基于单个彩色相机的新型全靶面、单相机三维数字图像相关(3D-DIC)方法。借助于设计巧妙的颜色分光光路,被测物体表面图像可以通过两条不同的光路达到相机靶面,采集的标定靶和实验件表面的彩色图像可以分离得到蓝色和红色子图像。通过使用3D-DIC分析标定靶和实验件表面分离后的蓝色和红色子图像,可以获得物体表面的三维形貌和变形。形貌测量、面内和离面平移、以及静动态三维变形实验验证了该单彩色相机3D-DIC方法的有效性和测量精准度。由于可避免双相机同步,且能实现无分辨率损失的全靶面三维形貌和变形测量,本文方法在需要实现瞬态位移和变形测量的爆炸、冲击、振动等领域中具有广阔重要的应用前景。     

Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements

This article reviews recently developed methods for constitutive parameter identification based on kinematic full-field measurements, namely the finite element model updating method (FEMU), the constitutive equation gap method (CEGM), the virtual fields method (VFM), the equilibrium gap method (EGM) and the reciprocity gap method (RGM). Their formulation and underlying principles are presented and discussed. These identification techniques are then applied to full-field experimental data obtained on four different experiments, namely (i) a tensile test, (ii) the Brazilian test, (iii) a shear-flexural test, and (iv) a biaxial test. Test (iv) features a non-uniform damage field, and hence non-uniform equivalent elastic properties, while tests (i), (ii) and (iii) deal with the identification of uniform anisotropic elastic properties. Tests (ii), (iii) and (iv) involve non-uniform strain fields in the region of interest. Working group “Identification” of the French CNRS research network (GDR 2519) “Mesures de champs et identification en Mécanique des Solides / Full-field Measurements and Identification in Solid Mechanics”.     

基于SGMD及LWOA-ELM的有限元模型修正

为得到待修正参数与结构响应之间的关系,提高模型修正的效率和精度,提出了一种基于辛几何模态分解(SGMD)和Lévy飞行鲸鱼优化算法(LWOA)优化极限学习机(ELM)的有限元模型修正(FEMU)方法。首先,对加速度频响函数(AFRF)进行SGMD分解,采用能量熵增量法确定重组辛几何分量(SGC)构成SGC矩阵。然后,利用LWOA对ELM的权值和阈值进行优化,提高ELM模型的预测效率,以LWOA-ELM为代理模型映射出待修正参数与SGC矩阵之间的关系。最后,以试验频响函数SGC矩阵与LWOA-ELM模型输出所得矩阵差值的F-范数最小为目标函数,结合LWOA求解待修正参数。算例分析表明,提出的方法用于有限元模型修正有较好的可行性和有效性。以SGC矩阵表征AFRF的修正方法,有较好的噪声鲁棒性;LWOA-ELM作为代理模型预测精度高,泛化能力强。     

Real-time Bayesian data assimilation with data selection,correction of model bias,and on-the-fly uncertainty propagation

The work introduces new advanced numerical tools for data assimilation in structural mechanics. Considering the general Bayesian inference context, the proposed approach performs real-time and robust sequential updating of selected parameters of a numerical model from noisy measurements, so that accurate predictions on outputs of interest can be made from the numerical simulator. The approach leans on the joint use of Transport Map sampling and PGD model reduction into the Bayesian framework. In addition, a procedure for the dynamical and data-based correction of model bias during the sequential Bayesian inference is set up, and a procedure based on sensitivity analysis is proposed for the selection of the most relevant data among a large set of data, as encountered for instance with full-field measurements coming from digital image/volume correlation (DIC/DVC) technologies. The performance of the overall numerical strategy is illustrated on a specific example addressing structural integrity on damageable concrete structures, and dealing with the prediction of crack propagation from a damage model and DIC experimental data.     

Follow-up of a panel restoration procedure through image correlation and finite element modeling

Residual stress estimation is an important question for structural integrity. Since residual stresses are self-balanced stress fields, a classical way to obtain information on them is to remove a part of the structure, and observe the structure displacement field arising from the stress redistribution. The hole-drilling method is such an approach. In some cases, as for the present one concerning a painted panel of cultural heritage, the hole-drilling method is not suited (a structure with a complex geometry, few tests allowed) but one can take advantage of structural modifications if they are monitored (here, a restoration act). We therefore describe in this article a model updating approach, focusing on the residual stress estimation and not on the material parameter identification.This study couples an optical non-invasive shape measurement (digital image correlation, using a projected speckle pattern on the painted panel, with luminance compensation) and a numerical approach (3D finite elements) for the model updating. The 3D stereo-correlation is used to measure a partial displacement field between three different states of the structure (at three different times of the restoration act). The numerical part concerns stress evaluation, once the model and the experiments are compared using a geometric mapping and a spatial projection of discrete fields. Using modeling and identification, the simulation is used to obtain the residual stresses in the panel, before and after the restoration.     

Full-Field Surface 3D Shape and Displacement Measurements Using an Unfocused Plenoptic Camera

Full-field surface 3D shape and displacement measurements using a single commercial unfocused plenoptic camera (Lytro Illum) are reported in this work. Before measurements, the unfocused plenoptic camera is calibrated with two consecutive steps, including lateral calibration and depth calibration. Each raw image of a checkerboard pattern recorded by Lytro Illum is first extracted to an array of sub-aperture images (SAIs), and the center sub-aperture images (CSAIs) at diverse poses are used for lateral calibration to determine intrinsic and extrinsic parameters. The parallax maps between the CSAI and the remaining SAIs at each pose are then determined for depth parameters estimation using depth calibration. Furthermore, a newly developed physical-based depth distortion model is established to correct the serious distortion of the depth field. To realize shape and deformation measurements, the raw images of a test sample with speckle patterns premade on its surface are captured by Lytro Illum and extracted to arrays of SAIs. The parallax maps between the CSAI and the target SAIs are obtained using subset-based digital image correlation. Based on the pre-computed intrinsic and depth parameters and the disparity map, the full-field surface 3D shape and displacement of a test object are finally determined. The effectiveness and accuracy of the proposed approach are evaluated by a set of experiments involving the shape reconstruction of a cylinder, in-plane and out-of-plane displacement measurements of a flat plate and 3D full-field displacement measurements of a cantilever beam. The preliminary results indicate that the proposed method is expected to become a novel approach for full-field surface 3D shape and displacement measurements.     

基于贝叶斯信息融合的岸桥有限元模型修正研究

为建立精确的岸桥有限元模型,研究了基于贝叶斯信息融合的模型修正方法．通过方差分析,确定待修正参数,利用中心复合试验设计获取样本点,根据有限元计算结果与实测的结果残差为目标函数获得响应样本．拟合样本点和响应样本值构建二阶多项式响应面模型,并检验响应面模型的精度．基于贝叶斯理论更新融合系数来优化响应面参数,从而获得修正模型．以宁波大榭3号岸桥为工程背景,对比修正后的模态频率和实测频率,最大频率相对误差不超过5%,进而验证了基于贝叶斯信息融合的动力学有限元模型修正方法的有效性．修正后的有限元模型可进一步应用于岸桥的健康监测和安全评估．     

Diffraction Assisted Image Correlation: A Novel Method for Measuring Three-Dimensional Deformation using Two-Dimensional Digital Image Correlation

Digital Image Correlation (DIC) provides a full-field non-contact optical method for accurate deformation measurement of materials, devices and structures. The measurement of three-dimensional (3D) deformation using DIC in general requires imaging with two cameras and a 3D-DIC code. In the present work, a new experimental technique, namely, Diffraction Assisted Image Correlation (DAIC) for 3D displacement measurement using a single camera and 2D-DIC algorithm is presented. A transmission diffraction grating is placed between the specimen and the camera, resulting in multiple images which are then used to obtain apparent in-plane displacements using 2D-DIC. The true in-plane and out-of-plane displacements of the specimen are obtained from the apparent in-plane displacements and the diffraction angle of the grating. The validity and accuracy of the DAIC method are demonstrated through 3D displacement measurement of a small thin membrane. This technique provides new avenues for performing 3D deformation measurements at small length scales and/or dynamic loading conditions.     

Full-field characterization of mechanical behavior of polyurethane foams

The foam material of interest in this investigation is a rigid closed-cell polyurethane foam PMDI with a nominal density of 20 pcf (320 kg/m³). Three separate types of compression experiments were conducted on foam specimens. The heterogeneous deformation of foam specimens and strain concentration at the foam–steel interface were obtained using the 3-dimensional digital image correlation (3D-DIC) technique. These experiments demonstrated that the 3D-DIC technique is able to obtain accurate and full-field large deformation of foam specimens, including strain concentrations. The experiments also showed the effects of loading configurations on deformation and strain concentration in foam specimens. These DIC results provided experimental data to validate the previously developed viscoplastic foam model (VFM). In the first experiment, cubic foam specimens were compressed uniaxially up to 60%. The full-field surface displacement and strain distributions obtained using the 3D-DIC technique provided detailed information about the inhomogeneous deformation over the area of interest during compression. In the second experiment, compression tests were conducted for cubic foam specimens with a steel cylinder inclusion, which imitate the deformation of foam components in a package under crush conditions. The strain concentration at the interface between the steel cylinder and the foam specimen was studied in detail. In the third experiment, the foam specimens were loaded by a steel cylinder passing through the center of the specimens rather than from its end surface, which created a loading condition of the foam components similar to a package that has been dropped. To study the effects of confinement, the strain concentration and displacement distribution over the defined sections were compared for cases with and without a confinement fixture.     

高速3D-DIC测试技术在装甲钢贯穿试验中的应用

数字图像相关（digital image correlation, DIC）技术作为一种非接触、非干涉的全场无损光学量测技术,可获取材料表面的动态变形信息和破坏过程。为了评估装甲钢的抗弹性能并探索高速三维数字图像相关（3D-DIC）技术在钢板贯穿试验测试中的应用,基于氢氧爆轰驱动弹道枪开展了7发15 mm口径可变形弹体以不同速度（255～568 m/s）冲击不同厚度（5、8和10 mm）高强高硬装甲钢板的试验,并结合帧率为144 000 s?1的高速3D-DIC测试技术获取了靶板的离面位移和应变时程。随后,基于前期标定并验证的装甲钢本构模型参数,对上述试验进行了数值模拟。通过对比弹体残余速度和长度验证了有限元分析方法的可靠性。进一步通过对比试验与数值模拟得到的靶背离面位移时程曲线和不同时刻靶背的应变云图,验证了高速3D-DIC测试结果的准确性。最后,对比分析了靶板最大离面位移与弹体冲击速度和装甲钢板厚度的关系。高速3D-DIC测试技术的应用可为相关试验测试提供参考,靶板最大离面位移分析结果可为屏障类防护结构的分析验证和优化设计提供试验依据。     

一种结构参数识别的两阶段方法

针对测量信息不完备的剪切型结构 ,建立了一种两阶段系统识别的复合反演方法 ,这种方法包括两部分 :子结构地震动反演和结构参数识别。首先 ,选取可观测的子结构 ,利用一维地震动作用于结构的力学特性 ,将子结构动力方程的有限元列式进行变换 ,得到适合于最小二乘法的简单形式 ,解决了测量信息不完备及结构参数未知条件下的地震动反演问题。其次 ,根据子结构反演得到的地震动输入 ,采用结构参数时域识别技术中的加权整体迭代 -广义卡尔曼滤波器方法 ,成功地识别出了有限测量条件下单元水平结构参数     

Sequential non-linear least-square estimation for damage identification of structures with unknown inputs and unknown outputs

The detection of structural damages real-time on-line, based on vibration data measured from sensors, is an important but challenging research topic, and it has received considerable attentions recently. Due to practical limitations, it is highly desirable to install as few sensors as possible in the structural health monitoring system, leading to incomplete measurements of structural responses and excitations. The traditional time-domain analysis techniques, such as the least-square estimation (LSE) method and the extended Kalman filter (EKF) approach, require that all the external excitations (inputs) be available, which may not be the case for most structural health monitoring systems. Recently, the adaptive sequential non-linear least-square estimate (SNLSE) method has been proposed for the on-line identification of structural damages. In this paper, we extend the SNLSE method to cover the general case with unknown (unmeasured) excitations (inputs) and unknown (unmeasured) acceleration responses (outputs) in order to reduce the number of sensors required in the structural health monitoring system, referred to as the SNLSE-UI-UO. Analytic recursive solutions for the new approach are derived and presented. The accuracy and effectiveness of the proposed approach have been demonstrated using the Phase I ASCE structural health monitoring benchmark building, a 5-degree-of-freedom non-linear hysteretic building model, and a 3-story steel frame finite-element model. Simulation results indicate that the proposed approach is capable of tracking the changes of structural parameters leading to the identification of damages.     

An Out-of-Plane Motion Compensation Strategy for Improving Material Parameter Estimation Accuracy with 2D Field Measurements

In-plane surface displacements, when measured with 2D Digital Image Correlation (2D-DIC), are very sensitive to out-of-plane displacement components. Any out-of-plane motion of the surface can pollute the measured field by introducing artificial displacements. These displacements are difficult to separate from the underlying response of the surface and thereby limit the application of 2D-DIC in inverse problems where the test specimen has significant motion in the out-of-plane direction. In the context of inverse problems, we propose to partially relax this condition of no out-of-plane motion in 2D-DIC. With this approach, only the out-of-plane rigid-body motion of the specimen surface, which is initially in-plane, needs to be avoided while the requirement of surface deformations to be primarily in-plane is essentially waived. Compensation, based on the pinhole camera model, for out-of-plane displacements of the surface in response to applied load is included within the error function of the minimization problem. The improvements in material parameter estimation, obtained by using the proposed compensation strategy, are demonstrated by an example. The proposed technique makes it possible to utilize 2D-DIC with a simple conventional lens for an increased number of inverse problems; and in the process avoiding the computational and experimental difficulties associated with 3D measurement methods as well as the high cost and magnification limitations of a telecentric lens.     

多相机3D-DIC及其在高温变形测量中的应用

将基于双目视觉的三维数字图像相关方法 (Three-dimension digital image correlation,3D-DIC)与多相机同步采集系统相结合,形成基于多相机的3D-DIC系统。依据三维空间误差(Three-dimensional residual,3Dresidual)最小原则,确定各点对应的最佳双目视觉系统,获得物体全场三维变形。以四相机3D-DIC系统为例,与测量精度达10~20nm的电子散斑干涉测量系统同时对平板的离面位移进行测量,并对测量得到的离面位移最大值进行了对比分析。结果显示,荷载较小时,四相机3D-DIC与电子散斑干涉测量系统误差稍大,最大达到2.7%;荷载增大,物体变形增大时,两种测量系统结果基本相同。文中讨论了四相机测量系统的不稳定对实验结果的影响。利用该四相机3D-DIC系统对镍合金不锈钢材料在高温场中的变形进行测量,获得了物体的三维变形场,并分析了材料的膨胀系数,得到了试件的热应变-温度曲线和膨胀系数随温度变化的关系式。     

用于物体表面形貌和变形测量的三维数字图像相关方法

使用单个摄像机的二维数字图像相关方法通常仅局限于平面物体的面内变形测量,而使用两个摄像机基于双目立体视觉原理的三维数字图像相关方法克服了这一局限,可对平面和曲面物体表面的三维形貌和载荷作用下的三维变形进行测量。本文介绍了三维数字图像相关方法的基本原理及其关键技术,并用两个典型的实验验证了该方法的有效性。     

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.     

Finite element model updating using variable separation

In the field of structural dynamics, reliable finite element response predictions are becoming increasingly important to industry and there is a genuine interest to improve these in the light of measured frequency response functions. Unlike modal-based model updating formulations, response-based methods have been applied only with limited success due to incomplete measurements and numerical ill-conditioning problems. The least squares approximation method is one of the methods used but often poses a problem of pseudo inverse due to the number of incomplete measurements. The proposed algorithm is a modification and extension of a previously-developed nonlinear least squares method for damage detection and finite element model updating. The paper derives explicit expressions for the first and second order partial derivatives with respect to the correction parameters and for the Jacobian matrix used in the Newton–Raphson solution of the nonlinear set of equations in order to avoid the pseudo inverse and to build a symmetrical system. The proposed method, assigned to a frequency parameterization which considers the minimum distance to be minimized, shows a good numerical stability. The performance of the method in localizing structural damage and updating model is examined using simulated measurements.