Accurate Displacement Research Articles

Multi-cylinder leveling control systems based on dual-valve parallel and adaptive eccentric torque suppression

In the composite material hydraulic press, the mismatched velocities between the movable beam and the multiple leveling cylinders produce disturbing superfluous forces, and the eccentric torque causes the movable beam to tilt when pressed. They severely damage the leveling displacement accuracy and limit the implementation of muti-cylinder system in higher precision field. A dual-loop leveling control strategy is proposed, comprising a dual-valve parallel pressure inner loop and an adaptive control displacement outer loop. Firstly, a dual-valve parallel scheme is proposed in the pressure inner loop, where a compensation valve is added in parallel with the original single-valve. A variable compensation valve spool algorithm is designed, considering both velocity and displacement to mitigate the effects of superfluous forces and achieve precise and smooth leveling. Secondly, a control strategy for the adaptive displacement outer loop is designed to estimate and compensate for eccentric torque. An innovative torque decoupling algorithm is formulated to overcome the challenge of indeterminate coupling relations between inner and outer loops caused by the adaptive incorporation of dual-loop control. Then, eccentric load compensation torque is decoupled to the multiple leveling cylinders and derives the desired pressure for the inner loop. The inner loop suppresses the disturbance of eccentric torque to enhance robust leveling precision. Finally, the effectiveness of the proposed strategy was validated through experimentation on the constructed hydraulic press leveling system test bench. The control strategy presented in this paper provides a reference for achieving smooth and precise control in multi-cylinder systems.

Bayesian model averaging and Bayesian inference-based probabilistic inversion method for arch dam zonal material parameters

Inversion analysis based on structural monitoring data is an important part of evaluating the working behaviour of structures. In this paper, a probabilistic inversion method for the elastic modulus of concrete in arch dams based on Bayesian model averaging (BMA) and Bayesian inference is proposed. According to the measured displacement data of an arch dam project, the influence of the separation accuracy of water pressure component, the inversion method and the selection of displacement measuring points on the inversion results of the elastic modulus of the dam body is studied. Example analysis results show that the accuracy of the multi-measurement point displacement and hydraulic pressure component separation model based on BMA-HST (Hydrostatic-seasonal-time) is at least 18.97 % higher than that of the single measurement point, and the relative error of the elastic modulus value of the multi-measurement point zonal inversion based on the Polynomial chaos-Kriging (PCK) proxy model and the Bayesian inference is only 6 % at the maximum, which indicates that the inversion model described in this paper is able to realize the high-precision inversion analysis of the material parameters of the zonal analysis of concrete dams at a relatively low computational cost.

In vivo Multi-perspective 3D + t Ultrasound Imaging and Motion Estimation of Abdominal Aortic Aneurysms.

Time-resolved three-dimensional ultrasound (3D + t US) is a promising imaging modality for monitoring abdominal aortic aneurysms (AAAs), providing their 3D geometry and motion. The lateral contrast of US is poor, a well-documented drawback which multi-perspective (MP) imaging could resolve. This study aims to show the feasibility of in vivo multi-perspective 3D + t ultrasound imaging of AAAs for improving the image contrast and displacement accuracy. To achieve this, single-perspective (SP) aortic ultrasound images from three different angles were spatiotemporally registered and fused, and the displacements were compounded. The fused MP had a significantly higher wall-lumen contrast than the SP images, for both patients and volunteers (P < .001). MP radial displacements patterns are smoother than SP patterns in 67% of volunteers and 92% of patients. The MP images from three angles have a decreased tracking error (P < .001 for all participants), and an improved SNRe compared to two out of three SP images (P < .05). This study has shown the added value of MP 3D + t US, improving both image contrast and displacement accuracy in AAA imaging. This is a step toward using multiple or large transducers in the clinic to capture the 3D geometry and strain more accurately, for patient-specific characterization of AAAs.

Application of model predictive control scheme for piezoelectric ceramic micro-displacement

Abstract Currently, in the domain of micro-control technology, many operations need to meet the standard of nanometer-level positioning. As a high-precision micro-displacement element, piezoelectric ceramics are widely used in the field of micro-control due to their characteristics such as no need for transmission devices, high resolution, large output force, small size, and fast response speed. However, the inherent hysteresis nonlinearity of piezoelectric ceramics can lead to decreased displacement accuracy, oscillation, and even system instability. Therefore, eliminating the influence of nonlinear characteristics and improving displacement accuracy possess significant practical importance. In this paper, a model predictive control scheme is designed, which achieves faster response speed and higher accuracy compared to traditional control methods such as Proportional Integral Derivative control. Relevant simulations and tests have been conducted. In this scheme, hysteresis nonlinearity is first modeled and then compensated through feedforward, thereby simplifying the displacement control of piezoelectric ceramics into a more manageable linear problem. Subsequently, a linear model predictive control method is adopted to control the entire system. Both simulation and experimental results have verified the effectiveness of this control method.

Sub‐Pixel Displacement Estimation With Deep Learning: Application to Optical Satellite Images Containing Sharp Displacements

AbstractOptical image correlation is a powerful method for remotely constraining ground movement from optical satellite imagery related to natural disasters (e.g., earthquakes, volcanoes, landslides). This approach enables the characterization, and identification of the causal factors and mechanisms underlying such processes. By employing sub‐pixel correlation algorithms, one can obtain highly accurate (m‐to‐cm level) displacement fields at high spatial resolution (dm‐to‐cm) by comparing satellite images acquired before and after a period of movement. However, this method generally assumes a homogeneous translation of all pixels within a given correlation window, which will lead to biased estimates of ground displacement if the real case is not well represented by such a simplification, especially when resolving ground displacements next to sharp gradients in displacement, such as those found in the near‐field of earthquake surface ruptures. In this paper, we present an innovative deep learning method estimating sub‐pixel displacement maps from optical satellite images for the retrieval of ground displacement. From the generation of a realistic simulated database, comprising Landsat‐8 satellite image pairs containing simulated sub‐pixel shifts and sharp discontinuities, we develop a Convolutional Neural Network able to retrieve sub‐pixel displacements. The comparison to state‐of‐the‐art correlation methods shows that our pipeline significantly reduces by 32% the estimation bias around fault ruptures, leading to more accurate characterization of the near‐field strain in surface rupturing earthquakes. Application of our model to the 2019 Ridgecrest earthquake demonstrates the ability of our model to accurately and quickly resolve ground displacement using real satellite images. Code is made available at https://gricad‐gitlab.univ‐grenoble‐alpes.fr/montagtr/cnn4l‐discontinuities.

Efficient Bayesian updating for deformation prediction of high rock slopes induced by excavation with monitoring data

This study develops an efficient Bayesian updating method with monitoring data for predicting the deformation of high rock slopes induced by excavation. The importance ranking based on random forest is introduced to identify the key rock parameters as random variables in the Bayesian updating. The surrogate models using support vector machine are constructed to approximate the physical numerical models using FLAC3D for evaluating slope deformation. A practical example involving deformation prediction of the excavated left-bank rock slope for the well-known Baihetan hydropower station in southwest China is presented. The results indicate that the developed Bayesian updating method can efficiently and accurately update the posterior distributions of rock parameters and predict the deformation of high rock slopes induced by excavation. Incorporating the monitoring data of displacement into the Bayesian updating can effectively reduce the uncertainty of rock parameters and displacement prediction. As a result, the displacement predictions made by the Bayesian updating are closer to the monitoring data than the prior displacement predictions. In addition, incorporating more monitoring data of displacement from the previous excavation stages produces more accurate displacement predictions for subsequent excavation stages.

Accelerometer-assisted computer vision data fusion framework for structural dynamic displacement reconstruction

Accurate displacement measurement provides crucial information for evaluating the structural health condition. It is important to note that both direct and indirect displacement measurement methods have their own limitations, which can be remedied by leveraging each other’s strengths. In this study, a novel accelerometer-assisted computer vision data fusion framework is developed for accurately reconstructing structural dynamic displacement. The framework does not require a specific mathematical model or prior knowledge about the data’s characteristics, allowing it to be applied more broadly across different applications without the constraints of model assumptions. The core of this framework is to integrate successive variational mode decomposition (SVMD) and Bayesian optimization approaches to adaptively determine the weight factors of the fusion components. Importantly, an enhanced optical flow approach is presented for converting pixel movement to structural displacement from natural targets. This approach can effectively reduce the selection of mis-matched points within the ROI (Region of Interest), thereby reducing drift errors. The developed framework is verified via shaking table tests of a reinforced concrete frame structure under seismic excitation. Results indicate that the developed framework excels in accurately estimating structural dynamic displacement. Compared to single-vision displacement identification results, the proposed framework demonstrates lower peak error (< 1.65 mm) and normalized root mean square error (< 0.30). Meanwhile, the reconstructed displacement, by introducing dynamic displacement component from acceleration measurement, presents a wider frequency range than the vision-based displacement.

Effect of random variation in input parameter on cracked orthotropic plate using extended isogeometric analysis (XIGA) under thermomechanical loading

Abstract This research introduces the Stochastic Extended Isogeometric Analysis (XIGA) method to investigate fracture behavior of isotropic and orthotropic materials under mechanical, thermal, and thermomechanical loads. Employing knot spans from Isogeometric Analysis (IGA) for domain discretization, the study utilizes identical basis functions for geometry construction and solution discretization. Utilizing Extended Finite Element Method (XFEM) enrichment functions, accurate crack face displacement discontinuity and tip singularity within the stress field are characterized. Additionally, employing a second-order perturbation technique within XIGA framework, the research derives mean and coefficient of variance values for mixed-mode Stress Intensity Factors (SIF). Stochastic variations in material elastic properties, crack length, and crack angle are considered in this computation. Credibility and robustness of the study are confirmed through comparative analyses against available literatures and Monte Carlo Simulations (MCS). The observed exceptional agreement validates the precision and reliability of the proposed stochastic XIGA method for fracture analysis in orthotropic material systems under thermomechanical loading conditions.

Design and Performance Evaluation of a Novel Vascular Interventional Surgery Robot

Introduction: Minimally invasive interventional surgical robots are used as delivery tools to assist healthcare professionals with medical devices such as micro guidewires and microcatheters during surgical treatment of cardiovascular and cerebrovascular obstruction, thus reducing surgical fatigue and radiation for healthcare professionals. Method: In this paper, a new vascular interventional surgical robot with multiple degrees of freedom is designed, its mechanism principle and movement mode is discussed in detail, and a prototype prototype is fabricated to make it simulate and reproduce the doctor's hand movements. Results: Through Abaqus simulation and platform experiments, the performance of the surgical robot is evaluated and analyzed from the perspectives of micro guidewire delivery displacement accuracy and radial clamping force for its high accuracy, high reliability, and safety. Conclusion: The results demonstrate that its surgical robot meets the requirements of providing vessel wall overload protection while the microcatheter delivery positioning accuracy error is less than 1 mm.

A novel shape sensing approach based on the coupling of Modal Virtual Sensor Expansion and iFEM: Numerical and experimental assessment on composite stiffened structures

Shape sensing, i.e. the reconstruction of the displacement field of a structure from discrete strain measurements, is becoming crucial for the development of a modern Structural Health Monitoring framework. Nevertheless, an obstacle to the affirmation of shape sensing as an efficient monitoring system for existing structures is represented by its requirement for a significant amount of sensors. Two shape sensing methods have proven to exhibit complementary characteristics in terms of accuracy and required sensors that make them suitable for different applications, the inverse Finite Element Method (iFEM) and the Modal Method (MM). In this work, the formulations of these two methods are coupled to obtain an accurate shape sensing approach that only requires a few strain sensors. In the proposed procedure, the MM is used to virtually expand the strains coming from a reduced number of strain measurement locations. The expanded set of strains is then used to perform the shape sensing with the iFEM. The proposed approach is numerically and experimentally tested on the displacement reconstruction of composite stiffened structures. The results of these analyses show that the formulation is able to strongly reduce the number of required sensors for the iFEM and achieve an extremely accurate displacement reconstruction.

A vector‐valued ground motion intensity measure for base‐isolated buildings in far‐field regions

AbstractIn this study, the effects of the spectral acceleration at the superstructure first‐mode period on the isolator displacement are investigated for far‐field ground motions. For this purpose, two different base‐isolated models are subjected to 165 far‐field ground motions. It is demonstrated that considering the spectral acceleration at the superstructure first‐mode period, besides that at the effective period, improves the estimation accuracy of isolator displacement. ASCE 7‐22 modifies the scaling period range to consider the superstructure first mode period and proposes the new period range from the superstructure first‐mode period to the 1.25 times effective period. In the ASCE 7‐22, the same weight factor is used for the whole period range. However, the present study shows that adding the superstructure first‐mode related period range with appropriate weight factor to the effective period‐based scaling range decreases the dispersion of isolator displacement in the nonlinear response history analyses (NRH). Then, to overcome the spectral shape effects on the fragility curves, a vector‐valued intensity measure parameter is obtained by combining spectral acceleration at the effective period and reduced spectral acceleration at the superstructure first‐mode period. The optimum contribution factor for the spectral acceleration at the superstructure first‐mode period is defined as the ratio of the superstructure first‐mode period to the effective period. The article shows that the proposed parameter is efficient and sufficient to be used as an intensity measure for far‐field ground motions. Furthermore, regression analysis results indicate that this vector‐valued intensity measure parameter correlates well with the isolator displacement. Further, the article shows that using the proposed IM parameter in the fragility curves makes the collapse margin ratio of these curves less sensitive to the spectral shape of the selected ground motions.

The forecasting of surface displacement for tunnel slopes utilizing the WD-IPSO-GRU model

To quickly assess slope stability based on field displacement monitoring data, this paper constructs a hybrid optimization model that predicts surface displacement during tunnel excavation in base-overburden slopes. The model combines Wavelet Decomposition (WD) with a Gated Recurrent Unit (GRU), and the GRU's hyperparameters are optimized using an Improved Particle Swarm Optimization algorithm (IPSO). The specific steps are as follows: First, the Wavelet Decomposition (WD) technique is applied to decompose the raw displacement data, extracting features at different time–frequency scales. Next, the Dropout technique is incorporated into the GRU model to prevent overfitting. Additionally, nonlinear inertia weight ω improved cognitive factor c1, and social factor c2 are introduced. The PSO algorithm is improved by integrating crossover and mutation concepts from genetic algorithms. Finally, the IPSO is used to optimize the number of neural units hN, HN, LN and dropout rates D1 and D2 in the GRU network architecture. After constructing the WD-IPSO-GRU model, a comprehensive comparison is made with various swarm intelligence algorithms and state-of-the-art models. The experimental results demonstrate that the WD-IPSO-GRU model significantly improves the prediction accuracy of surface displacement in slopes during tunnel excavation. Compared to directly using raw data for prediction, the introduction of the WD preprocessing technique improved the prediction accuracy at measurement points 01 and 02 by 28% and 45.9%, respectively. Additionally, with the model optimized by IPSO, the prediction accuracy at measurement points 01 and 02 increased by 76% and 56.7%, respectively. The WD-IPSO-GRU model effectively addresses the challenges of extracting features from univariate displacement time-series data and determining the parameters of the GRU network. It improves the prediction accuracy of surface displacement in base-overburden type slopes and demonstrates excellent generalization ability and reliability. The research results validate the potential application of the model in geotechnical engineering and provide strong support for assessing slope stability during tunnel excavation.

Far-Field Phase-Shifting Structured Light Illumination Enabled by Polarization Multiplexing Metasurface for Super-Resolution Imaging.

The phase-shifting structured light illumination technique is widely used in imaging but often relies on mechanical translation stages or spatial light modulators, leading to system instability, low displacement accuracy, and limited integration feasibility. In response to these challenges, we propose and demonstrate an approach for generating far-field phase-shifting structured light using a polarization multiplexing metasurface. By controlling the polarization states of incident and transmitted light, the metasurface creates a three-step displacement of structured light, eliminating the need to move samples or illumination sources. As a proof of concept, we experimentally demonstrate microscopic imaging using structured light illumination generated by metasurfaces, extracting high-frequency information from objects, and surpassing the diffraction limit. The proposed metasurface platform offers a promising approach for developing compact and robust phase-shifting imaging systems, with broad prospects in quantitative detection, machine vision, and beyond.

Visual Recognition Method for Lateral Swing of the Tail Rope

The large lateral displacement of tail ropes increases the risk of their wear and fracture, posing hidden dangers to the safety of the hoisting system. However, no suitable method is available to recognize the lateral swing of tail ropes online. A target-free visual measurement method, which includes the dual-branch SiamSeg, was proposed in this study. Considering the slender characteristics of tail ropes, the receptive field of the feature extraction network was enhanced via the Receptive Field Module (RFM), thereby strengthening the discriminability and integrity of tail rope features. The consistency loss constraints were added to the segmentation loss function to maximize the time sequence information of the video and further improve the accuracy of pixel-level displacement. Compared with other methods, the proposed approach achieved better segmentation effects. Comparison results synchronously measured by sensors revealed the effectiveness of this method and its potential for practical underground applications.

Exponential time propagators for elastodynamics

We propose a computationally efficient and systematically convergent approach for elastodynamics simulations. We recast the second-order dynamical equation of elastodynamics into an equivalent first-order system of coupled equations, so as to express the solution in the form of a Magnus expansion. With any spatial discretization, it entails computing the exponential of a matrix acting upon a vector. We employ an adaptive Krylov subspace approach to inexpensively and accurately evaluate the action of the exponential matrix on a vector. In particular, we use an apriori error estimate to predict the optimal Krylov subspace size required for each time-step size. We show that the Magnus expansion truncated after its first term provides quadratic and superquadratic convergence in the time-step for nonlinear and linear elastodynamics, respectively. We demonstrate the accuracy and efficiency of the proposed method for one linear (linear cantilever beam) and three nonlinear (nonlinear cantilever beam, soft tissue elastomer, and hyperelastic rubber) benchmark systems. For a desired accuracy in energy, displacement, and velocity, our method allows for 10−100× larger time-steps than conventional time-marching schemes such as Newmark-β method. Computationally, it translates to a ∼1000× and ∼10−100× speed-up over conventional time-marching schemes for linear and nonlinear elastodynamics, respectively.

Nonlinearity-suppressed micro-probe fiber optic interferometer for accurate long-range displacement measurements

Fiber-optic interferometry technology has advanced rapidly in recent years. However, accurate measurement of the displacement over long ranges remains a challenge. This study reveals that fiber interferometers with Fabry–Perot cavities experience non-elliptic multi-order nonlinear errors due to parasitic interference, leading to nonlinearity correction failures in the case of reflectivity mismatch. We proposes a nonlinearity-suppressed microprobe fiber interferometer to measure long-range displacement with high accuracy. The proposed microprobe interferometer structurally avoids non-elliptic nonlinearities from higher-order harmonics and ensures the effectiveness of the nonlinearity correction method. Moreover, we optimized the micro-sensing probe parameters based on an established multimedia transmission interference model to further improve the measurement accuracy of long-range displacement. Nonlinear errors are reduced from 2.3 nm to 0.92 nm in the application range 0–700 mm. The findings of this study demonstrate that the proposed nonlinearity-suppressed microprobe fiber interferometer is suitable for sub-nanometer accuracy measurements over displacements of several hundred millimeters, especially in limited-space scenarios. It has great potential for industries seeking breakthroughs in long-range displacement measurement, such as robotics, medical devices and manufacturing.

Computationally-efficient locking-free isogeometric discretizations of geometrically nonlinear Kirchhoff–Love shells

Discretizations based on the Bubnov-Galerkin method and the isoparametric concept suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements were originally introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose CASs and CASns elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. CASs and CASns elements are interpolation-based assumed-strain locking treatments. The assumed strains have C0 continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. CASs elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. CASns elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, CASs and CASns elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of CASs and CASns elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that CASs and CASns elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

An optimized MOPSA algorithm for highly accurate deterministic lateral displacement microfluidic sorting chip design

Abstract Deterministic lateral displacement (DLD) sorting is a passive sorting technology. Due to the advantages of small volume and high accuracy, the DLD devices have been used in many applications. The critical diameter (Dc) value of the DLD device depends on the geometric structure of the internal post array. However, the existing empirical formula cannot match all types of post-arrays. Achieving the desired Dc value typically involves multiple iterative processes, leading to increased labor and time costs. In this paper, we propose an optimized trajectory prediction algorithm based on the original MOPSA, which considers the impact of fluidic pressure on particle trajectories. The method is to stretch the 2D physical field space into 3D, the particle expands from a 2D circle into a 3D ball on the pressure analysis. The net pressure on the particle is calculated, and the displacement of the particle is predicted by combining Newton’s second law and the displacement formula. A DLD device with Dc=10.6 μm was designed using this optimized algorithm and fabricated. It is verified by separation of 10 μm and 11 μm polystyrene particles. The test results were consistent with the simulation results. Compared with the empirical formula and the original MOPSA, this optimized algorithm can improve the prediction accuracy of particle trajectories in DLD devices.

Smart DIC: User-independent, accurate and precise DIC measurement with self-adaptively selected optimal calculation parameters

Existing subset-based digital image correlation (DIC) must rely on user-selected key calculation parameters (i.e., subset size and shape function) to proceed with displacement/deformation analysis. However, the lack of clear guidelines for selecting these parameters leads to varying choices among users, thus introducing artificial uncertainty in DIC measurements. Previous theoretical analyses and experimental studies revealed that optimal calculation parameters must account for both local speckle pattern quality and deformation at each required point. That means these key parameters should vary at each calculation point, posing a significant challenge for optimal parameter selection. To tackle this challenge, a novel Smart-DIC is proposed to achieve user-independent, accurate and precise displacement field measurements. This method comprehensively considers the local speckle pattern quality and local deformation at each calculation point, and gives the explicit formulas to determine optimal calculation parameters. Based on these optimal calculation parameters, Smart-DIC outputs accurate and precise displacement measurements without the need for users’ inputs. To validate its metrological performance, numerical tests were processed using data from DIC Challenge 1.0 and real images of tensile tests of a commercial AL-based alloy. The experimental results underscore the capacity of Smart-DIC to efficiently achieve accurate and precise displacement measurements across various scenarios by automatically selecting optimal subset sizes and shape functions.