Field Projection Method Research Articles

Simulation and experiment on obstacle avoidance control of concrete pump truck boom based on improved danger field and gradient projection method

Aiming at the shortcomings of the traditional robot obstacle avoidance algorithm applied directly to the leader–follower task transformation of concrete pump truck boom, an algorithm combined with improved danger field and improved gradient projection method for obstacle avoidance control of the boom is proposed. In the method, a joint limit avoidance function is used to avoid the angle overrun of the boom joints. And the danger field expression is modified to improve its engineering suitability. Moreover, a smoothing adjustment factor is introduced to improve the smoothness of the leader–follower task transformation of the boom. The influence of key control parameters on the safety and accuracy of boom movement is discussed through the simulation, and the reasonable ranges of the parameters are given. Furthermore, experiments are carried out to verify the effectiveness of the algorithm. The method provides a solution for obstacle avoidance control of construction machinery with multijoint series boom.

An inverse method to reconstruct crack-tip cohesive zone laws for fatigue by numerical field projection

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loadings, but the functional form of such phenomenological models does not have a strong micromechanical basis. While numerical techniques have been developed to extract the crack-tip cohesive zone law for monotonic fracture from experimental strain information, elucidating the unloading-reloading hysteresis representing accumulated and irreversible damage under fatigue cycling is still challenging. Here, we introduce a novel field projection method (FPM) to reconstruct the cohesive zone laws for steady-state fatigue crack growth from elastic strain field information at various loading and unloading stages within a single steady-state fatigue cycle. The method is based on the Maxwell-Betti’s reciprocal theorem, which considers the material response to be linear elastic during loading and unloading, albeit with a reciprocity gap to account for the inhomogeneous residual elastic strain accumulated at the end of each fatigue cycle. Through numerical experiments, we demonstrate that this FPM is capable of accurately extracting the crack-tip cohesive tractions and separations, along with the elusive unloading-reloading hysteresis, which constitute the cohesive zone law for fatigue crack growth. We discuss the errors and uncertainties associated with this inverse approach.

Atomic mixed-mode cohesive-zone dual constitutive laws of impurity-embrittled grain boundaries in polycrystalline solids via nanoscale field projection method

Atomic-scale mixed-mode intergranular fracture, featured by non-local non-linear discrete atomic debonding processes near a crack tip along a grain boundary (GB), is modeled with a cohesive zone in a continuum scale controlled by cohesive-zone dual constitutive relations, a balanced traction-separation relationship, i.e., a conventional cohesive-zone law (CZL), and an unbalanced traction (UT)-centerline displacement (CD) relationship. In order to bridge two different scales, we developed a nanoscale field projection method (nano-FPM) based on atomic-scale interaction J and M integrals, for which asymptotic anisotropic elastic fields near an interfacial crack with balanced and unbalanced crack-face tractions are used as probing fields of the CZL and UT-CD relationship, respectively. Cracking phenomena along a GB in nickel with segregated sulfur impurity atoms under mixed-mode loadings are simulated with molecular dynamics. Embrittlement by sulfur impurity atoms is quantitatively estimated with the mixed-mode CZL of the GBs in nickel via the nano-FPM developed in this study. UT-CD relationship, a special feature of atomic fracture, representing the micromechanical change of surface stress between GB and cracked surfaces, is also obtained by the nano-FPM. New functional relationships for CZL, UT-CD relationship, and decohesion potential obtained by the nano-FPM are proposed to facilitate the implementation of them into mesoscale or continuum-scale analysis on mixed-mode intergranular fracture.

Inductance Calculation of Coreless-Type Linear PM Machines Based on Analytical Field Projection and Coil Separation Method

The analysis of winding inductance in electric machinery is crucial for both design and control implementations. This article presents an accurate and computationally efficient method to calculate the winding inductance in a coreless-type permanent magnet linear synchronous machine (PMLSM). The analysis is based on the novel approach combining the analytical field projection and the coil separation method. After dividing the coil domain into three regions, a two-dimensional winding inductance in the front view is calculated by solving the field governing equations in each subdomain. Subsequently, the model is extended in three dimensions by projecting the side view of the coil onto the 2-D plane, and the interpolation method is involved in accounting for the end-winding inductance. As a result, the proposed model can save much computation time to evaluate the winding inductance without relying on the finite element (FE) method. In the end, the validity of the analytical model is verified by FE and experimental results.

Estimation of a mixed-mode cohesive law for an interface crack between dissimilar materials

In this paper, a mixed-mode cohesive law for an interface crack between epoxy and TR (transparent thermoplastic) resin is inversely estimated by the field projection method using numerical solutions and experimentally measured displacements. Displacements in a region far away from the crack tip are measured by digital image correlation technique. An inverse analysis, the field projection method formulated from the interaction J- and M-integrals with numerical auxiliary fields, is carried out to estimate a mixed-mode cohesive law for an interface crack between dissimilar materials. In the present approach, nonlinear deformations and damage near the crack tip are converted into the relationships of tractions and separations on crack surfaces behind the crack tip. The phase angle of mixed-mode singularities of the interface crack is also obtained from measured displacements in this study.

Predictive and Multirate Sensor-Based Planning Under Uncertainty

In this paper, a general formulation of a predictive and multirate (MR) reactive planning method for intelligent vehicles (IVs) is introduced. The method handles path planning and trajectory planning for IVs in dynamic environments with uncertainty, in which the kinodynamic vehicle constraints are also taken into account. It is based on the potential field projection method (PFP), which combines the classical potential field (PF) method with the MR Kalman filter estimation. PFP takes into account the future object trajectories and their associated uncertainties, which makes it different from other look-ahead approaches. Here, a new PF is included in the Lagrange–Euler formulation in a natural way, accounting for the vehicle dynamics. The resulting accelerations are translated into control inputs that are considered in the estimation process. This leads to the generation of a local trajectory in real time (RT) that fully meets the constraints imposed by the kinematic and dynamic models of the IV. The properties of the method are demonstrated by simulation with MATLAB and C++ applications. Very good performance and execution times are achieved, even in challenging situations. In a scenario with 100 obstacles, a local trajectory is obtained in less than 1 s, which is suitable for RT applications.

Energetic Mesh-to-Mesh Projection of Magnetic Fields With Respect to Nonlinear B-H Curves

Galerkin projection, which transfers field distribution between different meshes, has proven its utility for various applications of finite element analysis. In this framework, the authors propose a field projection method that considers nonlinear magnetic properties. The formulations given are based on the energy conservation between different meshes.

Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loading, but the model parameters are phenomenological and are not closely tied to the underlying micromechanics of the problem. In this paper, we will inversely extract the cohesive zone laws for fatigue crack growth in an elasto-plastic ductile solid using a field projection method (FPM), which projects the equivalent tractions and separations at the cohesive crack-tip from field information outside the process zone. In our small-scale yielding model, a single row of discrete voids is deployed directly ahead of a crack in an elasto-plastic medium subjected to cyclic mode I K-field loading. Damage accumulation under cyclic loading is captured by the growth of voids within the micro-voiding zone ahead of the crack, while the evolution of the cohesive zone law representing the micro-voiding zone is inversely extracted via the FPM. We show that the field-projected cohesive zone law captures the essential micromechanisms of fatigue crack growth in the ductile medium: from loading and unloading hysteresis caused by void growth and plastic hardening, to the softening damage locus associated with crack propagation via a void by void growth mechanism. The results demonstrate the effectiveness of the FPM in obtaining a micromechanics-based cohesive zone law in-place of phenomenological models, which opens the way for a unified treatment of fatigue crack problems.

그래핀의 모드 I 균열에 대한 분자동역학 해석으로부터 균열 선단 응집 법칙의 평가

In this paper, a novel approach to estimate cohesive laws for the mode I fracture of the graphene is presented by combining molecular dynamic simulations and an inverse algorithm based on field projection method and finite element method. The determination of crack-tip cohesive laws of the graphene based on continuum mechanics is a non-trivial inverse problem of finding unknown tractions and separations from atomic simulations. The displacements of molecular dynamic simulations in a region far away from the crack tip are transferred to finite element nodes by using moving least square approximation. Inverse analyses for extracting unknown cohesive tractions and separation behind the crack tip can be carried out by using conservation nature of the interaction J- and M-integrals with numerical auxiliary fields which are generated by systematically imposing uniform surface tractions element-by-element along the crack surfaces in finite element models. The preset method can be a very successful approach to extract crack-tip cohesive laws from molecular dynamic simulations as a scale bridging method.

Inverse estimation of cohesive zone laws from experimentally measured displacements for the quasi-static mode I fracture of PMMA

In this paper, an efficient hybrid procedure combining experimental measurements and an inverse algorithm called the field projection method (FPM) is presented to estimate cohesive zone laws for the quasi-static mode I fracture of PMMA. The procedure is made up of the measurement of displacements in a region far away from the crack tip using the digital image correlation technique, its transfer to finite element (FE) nodes from measurement locations using the moving least square approximation, and inverse analyses using numerical auxiliary fields and the conservation nature of the interaction J- and M-integrals. Since it is very difficult to measure the tractions and separations ahead of the crack tip where a highly nonlinear behavior of materials emerges at very small scales, the present method can be a very successful approach to extract the cohesive zone laws from experimentally measured displacements in a far-field region. In particular, the FPM using numerical auxiliary fields facilitates its use for determining the mesh-dependent cohesive zone laws in FE computations.

Inverse extraction of interfacial tractions from elastic and elasto-plastic far-fields by nonlinear field projection

Determining the tractions along a surface or interface from measurement data in the far-fields of nonlinear materials is a challenging inverse problem which has significant engineering and nanoscience applications. Previously, a field projection method was established to identify the crack-tip cohesive zone constitutive relations in an isotropic elastic solid (Hong and Kim, 2003. J. Mech. Phys. Solids 51, 1267). In this paper, the field projection method is further generalized to extracting the tractions along interfaces bounded by nonlinear materials, both with and without pre-existing cracks. The new formulation is based on Maxwell–Betti's reciprocal theorem with a reciprocity gap associated with nonlinear materials. We express the unknown normal and shear tractions along the interface in terms of the Fourier series, and use specially constructed analytical auxiliary fields in the reciprocal theorem to extract the unknown Fourier coefficients from far-field data; the reciprocity gap in the formulation is iteratively determined with a set of numerical algorithms. Our detailed numerical experiments demonstrate that this nonlinear field projection method (NFPM) is well-suited for extracting the interfacial tractions from the far-field data of any nonlinear elastic or elasto-plastic material with known constitutive laws. Applications of the NFPM to experiments and atomistic simulations are discussed.

Inverse extraction of cohesive zone laws by field projection method using numerical auxiliary fields

SUMMARYThe cohesive zone law relates the cohesive tractions with the cohesive separations within the fracture process zone of a material and is used to quantify the strength and toughness of the material. Determining the material's cohesive zone law, however, is a nontrivial inverse problem of finding unknown tractions and separations from measurement data. Previously, a field projection method was established to extract the cohesive zone laws from far‐field data using interaction J‐integrals between the physical field of interest and auxiliary analytical probing fields. Here, we extend the universality of the field projection method and its ease of numerical implementation by using numerical auxiliary fields. These numerical fields are generated by systematically imposing uniform surface tractions element‐by‐element along the crack faces in finite element models. Then, interaction J‐ and M‐integrals between these auxiliary probing fields and the measurement field are used to reconstruct the traction and separation relationship along the crack faces. The effectiveness of this method in extracting the cohesive zone law from measured displacements in the far‐field region is demonstrated through numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

Nanometer Scale Mechanical Behavior of Grain Boundaries

ABSTRACTA nonlinear field projection method has been developed to study nanometer scale mechanical properties of grain boundaries in nanocrystalline FCC metals. The nonlinear field projection is based on the principle of virtual work, for virtual variations of atomic positions in equilibrium through nonlocal interatomic interactions such as EAM potential interaction, to get field-projected subatomic-resolution traction distributions on various grain boundaries. The analyses show that the field projected traction produces periodic concentrated compression sites on the grain boundary, which act as crack trapping or dislocation nucleation sites. The field projection was also used to assess the nanometer scale failure processes of Cu Σ5 grain boundaries doped with Pb. It was revealed that the Pb dopants prevented the emission of dislocations by grain boundary slip and embrittles the grain boundary.

Cohesive-zone laws for void growth — I. Experimental field projection of crack-tip crazing in glassy polymers

A hybrid framework for inverse analysis of crack-tip cohesive-zone model is developed in this two-part paper to measure cohesive-zone laws of void growth in polymers by combining analytical, experimental, and numerical approaches. This paper focuses on experimental measurements of the cohesive-zone laws for two nonlinear fracture processes in glassy polymers, namely multiple crazing in crack-growth toughening of rubber-toughened high-impact polystyrene (HIPS) and crazing of steady-state crack growth in polymethylmethacrylate (PMMA) under a methanol environment. To this end, electronic speckle pattern interferometry (ESPI) is first applied to measure the crack-tip displacement fields surrounding the fracture process zones in these polymers. These fields are subsequently equilibrium smoothed and used in the extraction of the cohesive-zone laws via an analytical solution method of the inverse problem, the planar field projection method (P-FPM) [Hong, S., Kim, K.-S., 2003. Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method. Journal of the Mechanics and Physics of Solids 51, 1267–1286]. Results show that the proposed framework of the P-FPM could provide a systematic way of finding the shape of the cohesive-zone laws governed by the different micro-mechanisms in the fracture processes. In HIPS, inter-particle multiple crazing develops and the craze zone broadens ahead of a crack-tip under mechanical loading. The corresponding cohesive-zone relationship of the multiple-craze zone is found to be highly convex, which indicates effectiveness of rubber particle toughening. It is also observed that the effective peak traction, 7 MPa, in the crack-tip cohesive zone of HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to stress multi-axiality effects. In contrast, in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze-void initiation to about 9 MPa and the fibril pull-out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction–separation cohesive-zone relationship, signifies environmental embrittlement of PMMA. These experimentally determined cohesive-zone laws are compared with detailed numerical analyses of effective microscale-void growth ahead of a crack tip in Part II.

Cohesive zone laws for void growth — II. Numerical field projection of elasto-plastic fracture processes with vapor pressure

Modeling ductile fracture processes using Gurson-type cell elements has achieved considerable success in recent years. However, incorporating the full mechanisms of void growth and coalescence in cohesive zone laws for ductile fracture still remains an open challenge. In this work, a planar field projection method, combined with equilibrium field regularization, is used to extract crack-tip cohesive zone laws of void growth in an elastic–plastic solid. To this end, a single row of void-containing cell elements is deployed directly ahead of a crack in an elastic–plastic medium subjected to a remote K-field loading; the macroscopic behavior of each cell element is governed by the Gurson porous material relation, extended to incorporate vapor pressure effects. A thin elastic strip surrounding this fracture process zone is introduced, from which the cohesive zone variables can be extracted via the planar field projection method. We show that the material's initial porosity induces a highly convex traction–separation relationship — the cohesive traction reaches the peak almost instantaneously and decreases gradually with void growth, before succumbing to rapid softening during coalescence. The profile of this numerically extracted cohesive zone law is consistent with experimentally determined cohesive zone law in Part I for multiple micro-crazing in HIPS. In the presence of vapor pressure, both the cohesive traction and energy are dramatically lowered; the shape of the cohesive zone law, however, remains highly convex, which suggests that diffusive damage is still the governing failure mechanism.

Nanoscale planar field projections of atomic decohesion and slip in crystalline solids. Part I. A crack-tip cohesive zone

The field projection method of Hong and Kim (2003) to identify the crack-tip cohesive zone constitutive relations in an isotropic elastic solid is extended to a nanoscale planar field projection of a cohesive crack tip on an interface between two anisotropic solids. This formulation is applicable to the elastic field of a cohesive crack tip on an interface or in a homogeneous material for any combination of anisotropies. This method is based on a new orthogonal eigenfunction expansion of the elastic field around an interfacial cohesive crack, as well as on the effective use of interaction J integrals. The nanoscale planar field projection is applied to characterizing a crack-tip cohesive zone naturally arising in the fields of atomistics. The atomistic fields analyzed are obtained from molecular statics simulations of decohesion in a gold single crystal along a direction in a (111) plane, for which the interatomic interactions are described by an embedded atom method potential. The field projection provides cohesive traction, interface separation, and the surface-stress gradient caused by the gradual variation of surface formation within the cohesive zone. Therefore, the cohesive traction and surface energy gradient can be measured as functions of the cohesive zone displacements. The introduction of an atomistic hybrid reference configuration for the deformation analysis has made it possible to complete the field projection and to evaluate the energy release rate of decohesion with high precision. The results of the hybrid analyses of the atomistics and continuum show that there is a nanoscale mechanism of decohesion lattice trapping or hardening caused by the characteristics of non-local atomistic deformations near the crack tip. These characteristics are represented by surface relaxation and the development of surface stresses in the cohesive zone.

Nano and Micro Mechanical Measurement of Interaction Forces Between Solid Surfaces

Two different types of experimental methods have beeen developed for measuring lateral interaction forces between two solid surfaces for nano- and micro-meter scale contacts. One is the type of direct measurement methods which typically utilize AFM instrumentations. In the direct lateral force measurements some size-scale effects are commonly observed due to the effects of adhesion and surface roughness. A recent development of a fine AFM lateral force calibration method, a diamagnetic lateral force calibrator, has made it possible to study such size-scale effects systematically. The other type is the field projection method which requires a high resolution measurement of a deformation field near the edge of a contact. For such measurements a comprehensive map of deformation measurement techniques is introduced in a domain of spatial and strain resolutions. This technique provides a way of assessing the non-uniform distribution of the surface interaction forces for nano and micro-meter scale contacts.

Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method

A solution method of an inverse problem is developed to extract cohesive-zone laws from elastic far-fields surrounding a crack-tip cohesive zone. The solution method is named the “field projection method (FPM).” In the process of developing the method a general form of cohesive-crack-tip fields is obtained and used for eigenfunction expansions of the plane elastic field in a complex variable representation. The closing tractions and the separation-gradients at the cohesive zone are expressed in terms of orthogonal polynomial series expansions of the general-form complex functions. The series expansion forms a set of cohesive-crack-tip eigenfunctions, which is complete and orthogonal in the sense of the interaction J-integral in the far field as well as at the cohesive-zone faces. The coefficients of the eigenfunctions in the J-orthogonal representation are extracted directly, using interaction J-integrals in the far field between the physical field of interest and auxiliary probing fields. The path-independence of the interaction J-integral enables us to identify the cohesive-zone variables, i.e. tractions and separations, and thus the cohesive-zone constitutive laws uniquely from the far-field data. A set of numerical algorithms is developed for the inversion method and the results from numerical experiments suggest that the proposed algorithms are well suited for extracting cohesive-zone laws from the far-field data. The set includes methods to find the position and size of a cohesive zone. Further included are discussions on error analysis and stability of the inversion scheme.

An analogue model for teaching reflection and refraction of waves

A concrete model for teaching the concepts of reflection and refraction is described. This model is based on the use of wave strips which can be utilized by application of the principle of wave-front continuity to demonstrate the concept of reflection and Snell’s law. A single and split field projection method is described for use in the classroom to demonstrate wave-front matching at a boundary.