Multiple Point Constraints Research Articles

A Unit-Load Approach for Reliability-Based Design Optimization of Linear Structures under Random Loads and Boundary Conditions

The low order Taylor’s series expansion was employed in this study to estimate the reliability indices of the failure criteria for reliability-based design optimization of a linear static structure subjected to random loads and boundary conditions. By taking the advantage of the linear superposition principle, only a few analyses of the structure subjected to unit-loads are needed through the entire optimization process to produce acceptable results. Two structural examples are presented in this study to illustrate the effectiveness of the proposed approach for reliability-based design optimization: one deals with a truss structure subjected to random multiple point constraints, and the other conducts shape design optimization of a plane stress problem subjected to random point loads. Both examples were formulated and solved by the finite element method. The first example used the penalty method to reformulate the multiple point constraints as external loads, while the second example introduced an approach to propagate the uncertainty linearly from the nodal displacement vector to the nodal von Mises stress vector. The final designs obtained from the reliability-based design optimization were validated through Monte Carlo simulation. This validation process was completed with only four unit-load analyses for the first example and two for the second example.

Local–global DVC analyses confirm theoretical predictions for deformation and damage onset in torsion of pantographic metamaterial

Pantographs are a special class of metamaterials that do not obey theories of first gradient continua. In the present work, a pantographic metamaterial was 3D printed and subjected to in situ torsion. The experiment was designed in order to investigate damage onset modalities in large torsional deformations of pantographic mesostructures. To account for the presence of true pivots, multiple point constraints were implemented in the DVC procedure. Local–global DVC was validated thanks to a series of steps accounting for numerous pivots. It was possible to determine the dominant damage mechanism in torsion. Damage was detected thanks to the strain and residual fields. Moreover, important modeling indications were obtained to guide further theoretical and experimental investigations.

Lightweight Chassis Design of Hybrid Trucks Considering Multiple Road Conditions and Constraints

The paper describes a fully automated process to generate a shell-based finite element model of a large hybrid truck chassis to perform mass optimization considering multiple load cases and multiple constraints. A truck chassis consists of different parts that could be optimized using shape and size optimization. The cross members are represented by beams, and other components of the truck (batteries, engine, fuel tanks, etc.) are represented by appropriate point masses and are attached to the rail using multiple point constraints to create a mathematical model. Medium-fidelity finite element models are developed for front and rear suspensions and they are attached to the chassis using multiple point constraints, hence creating the finite element model of the complete truck. In the optimization problem, a set of five load conditions, each of which corresponds to a road event, is considered, and constraints are imposed on maximum allowable von Mises stress and the first vertical bending frequency. The structure is optimized by implementing the particle swarm optimization algorithm using parallel processing. A mass reduction of about 13.25% with respect to the baseline model is achieved.

A fast calibration method of the robot’s 3D vision system

Calibration accuracy and convenience are very important for the rapid production of vision-based robots. Although there are many calibration methods for the 3D vision system, the convenience of calibration cannot meet the needs of industrial sites. In this paper, we propose a fast calibration method based on a 3D calibration block. Its key point is to complete the calibration process through multiple space point constraints, and the space points used are jointly solved by multiple planes obtained by local point cloud fitting. The experimental results show that the proposed method can quickly complete the calibration of the robot’s 3D vision system, and the calibration accuracy can meet low precision occasions.

Data-driven streamline stiffener path optimization (SSPO) for sparse stiffener layout design of non-uniform curved grid-stiffened composite (NCGC) structures

The homogenization-based streamline stiffener path optimization (SSPO) method was previously proposed by the authors for stiffener layout design of non-uniform curved grid-stiffened composite (NCGC) structures. The applications are limited to densely distributed stiffeners with no significant scale effect in the homogenization process, which is revealed in this work. A data-driven SSPO method, which is inspired by the popular principal component analysis (PCA) in data processing, is proposed for sparse stiffener layout design. With several referential layouts as input, the PCA generates a sequence of principal layouts. Together with weighting factors and distortion techniques, using only a small number of dominant principal layouts is enough to represent most possible synthesized layouts, which significantly reduces the number of design variables while preserving and extending a large design space. This strategy is further enhanced with a detailed modeling technique based on multiple point constraints (MPCs) to automate the modeling process. The data-driven SSPO method can be integrated with either gradient-based or artificial intelligence (AI) algorithms due to the small number of design variables. A flexible objective target, to maximize the critical buckling load or minimize the structural weight, can be used since the structural responses are evaluated on the detailed model. Numerical examples demonstrate the effectiveness of the proposed method in optimizing the layout of sparsely distributed stiffeners.

Robust adaptive beamforming with mainlobe maintenance of monopulse beams

Adaptive monopulse approaches for angularly localising a main-beam target in jamming are usually based on variations of the concept of optimising the sum (Σ) beam via standard adaptation and then optimising the delta (Δ) beam subject to maintain the monopulse slope. These methods have some deficiencies: they do not address target self-nulling issues; they lack tapering, leading to high sidelobes; they introduce multiple point constraints, which may result in target leakage; and they assume linearity of the monopulse ratio curve, something valid only in a small region close to the beam steering direction. The method of this study addresses these and other concerns by maintaining tapered Σ and Δ beams and fully preserving their mainlobes while adaptively suppressing jammers. This approach provides the added bonus of enabling operating in the monopulse curve's non-linear region at wider beamwidths, hence also increasing the efficiency of radar searching and tracking functions. It is shown that such approach is extremely robust and flexible, as it avoids target self-nulling effects due to beam mispointing, it is immune to target and interference dithering effects, and it can be easily extended for super-resolving multiple main-beam targets.

Vertex assigned morphing for parameter free shape optimization of 3-dimensional solid structures

In the past the Vertex Morphing method has mainly been used in the context of optimization of thin walled shell structures or fluid mechanical problems. In this paper an extension of this parameter free shape optimization method to 3-dimensional solid structures based on the free Finite Element code CalculiX is presented. Key element of the whole process is an efficient adjoint sensitivity analysis that can handle many design variables providing gradient information for the response functions of interest, e.g. stresses and mass. Typically, parameter free shape optimal designs result in non-smooth geometries and suffer from mesh dependencies. To avoid these phenomena an intermediate filter operation in combination with several post processing steps of the raw sensitivities is added. In the optimization, the filter size is also used as an additional design parameter controlling the curvature of the optimal shape. In many engineering optimization applications constraints have to be considered. These restrictions are included in the optimization problem by the use of Rosen’s gradient projection method. For the successive shape update a pseudo linear-elastic static Finite Element analysis with multiple point constraints and a varying Young’s modulus is performed in every design iteration. Finally, the capabilities of the extended Vertex Morphing method are demonstrated for an academic example and a geometrically complex part from aerospace industry.

Determination of optimal measurement configurations for self-calibrating a robotic visual inspection system with multiple point constraints

In this paper, we propose an algorithm to determine optimal measurement configurations for self-calibrating a robotic visual inspection system with multiple point constraints. The algorithm aims to improve the robotic visual inspection system’s calibration accuracy. To do so, a pre-calibration of the robotic visual inspection system is needed to obtain the hand-eye and robot exterior relationship to implement the inverse kinematic algorithm. The candidate measurement configurations with one point constraint can be obtained using the inverse kinematic algorithm for the robotic visual inspection system, so DETMAX is implemented to determine a given number of optimal measurement configurations from the candidate measurement configurations. Particle swarm optimization is used to optimize the positions of the multiple points one by one. To verify the efficiency of the proposed approach, experiment evaluation is conducted on a robotic visual inspection system.

Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints

The wavelet methods have been extensively adopted and integrated in various numerical methods to solve partial differential equations. The wavelet functions, however, do not satisfy the Kronecker delta function properties, special treatment methods for imposing the Dirichlet-type boundary conditions are thus required. It motivates us to present in this paper a novel treatment technique for the essential boundary conditions (BCs) in the spline-based wavelet Galerkin method (WGM), taking the advantages of the multiple point constraints (MPCs) and adaptivity. The linear B-spline scaling function and multilevel wavelet functions are employed as basis functions. The effectiveness of the present method is addressed, and in particular the applicability of the MPCs is also investigated. In the proposed technique, MPC equations based on the tying relations of the wavelet basis functions along the essential BCs are developed. The stiffness matrix is degenerated based on the MPC equations to impose the BCs. The numerical implementation is simple, and no additional degrees of freedom are needed in the system of linear equations. The accuracy of the present formulation in treating the BCs in the WGM is high, which is illustrated through a number of representative numerical examples including an adaptive analysis.

Multiple-point constraint applications for the finite element analysis of shear deformable composite beams – Variational multiscale approach to enforce full composite action

Composite beams that consist of two or more shear deformable layers find widespread applications in a variety of engineering structures. In the computational modelling of composite beams, the layers can be stacked together and connected conveniently at the nodes by using multiple-point constraints. However, this type of modelling does not inherit the kinematic behaviour of the continuous case and thus full-interaction between the layers cannot be always imposed by applying multiple-point constraints at the nodes. The work herein shows that in multiple-point constraint applications full composite action between the shear deformable layers can be recovered by using the variational multiscale approach. The originality of this study is in the interpretation of the multiple-point constraint application as the solution in a superfluously extended space because of the weakening in the kinematic constraints. It is shown that full composite action between the beam layers can be recovered by excluding the identified fine-scale effect from the solution of the multiple point constraint application. Selected examples illustrate the effects of loading and relative layer stiffness on the numerical error as well as modelling options for fully composite and delaminated beams.

The extended unsymmetric frontal solution for multiple-point constraints

Purpose – The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling. Design/methodology/approach – Re-written frontal solution method with a priori pivot and front sequence. OpenMP parallelization, nearly linear (in elimination and substitution) up to 40 threads. Constraints enforced at the local assembling stage. Findings – When compared with both standard sparse solvers and classical frontal implementations, memory requirements and code size are significantly reduced. Research limitations/implications – Large, non-linear problems with constraints typically make use of the Newton method with Lagrange multipliers. In the context of the solution of problems with large number of constraints, the matrix transformation methods (MTM) are often more cost-effective. The paper presents a complete solution, with topological ordering, for this problem. Practical implications – A complete software package in Fortran 2003 is described. Examples of clique-based problems are shown with large systems solved in core. Social implications – More realistic non-linear problems can be solved with this Frontal code at the core of the Newton method. Originality/value – Use of topological ordering of constraints. A-priori pivot and front sequences. No need for symbolic assembling. Constraints treated at the core of the Frontal solver. Use of OpenMP in the main Frontal loop, now quantified. Availability of Software.

Coupling of heterogeneous kinematics and Finite Element approximations applied to composite beam structures

In the framework of the modeling of composite beam structures, the eXtended Variational Formulation (XVF) is carried out to couple different kinematics. The purpose is to take advantages of efficient models and reduce the overall computational cost without loss of local precision. In this way, the structure is divided into non-overlapping domains with different kinematics. Local domains of interest are described with advanced models, such as refined Sinus model, to precisely describe local behavior, while the remaining global domain uses simple classical models. Each of the kinematics needs a suitable Finite Element (FE) approximation, therefore the coupling of different FE approximations is also addressed. The present approach is assessed on homogeneous and sandwich structures. It is compared with classical Multiple Point Constraints vias penalty. The study has shown the need of the introduction of a new operator for the layered structures. The results are very promising.

Implicit solutions with consistent additive and multiplicative components

This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative constituents with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudo-elements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture.

Variational multiscale approach to enforce perfect bond in multiple-point constraint applications when forming composite beams

Composite laminates that consist of two or more layers find widespread applications in a variety of engineering structures. In the computational modelling of composite laminates, the layers can be stacked together and connected conveniently at the nodes by using multiple-point constraints (MPCs). However, this type of modelling leads to weakening of the kinematic constraint conditions imposed by the bond between the juxtaposed layers and as a consequence, MPCs application at the nodes produces behaviour that is softer than the perfectly bonded composite beam behaviour. The work herein shows that when kinematic conditions for composite action are weakly imposed in the variational form, they can be enforced in the point-wise sense by proper selection of the interpolation field or otherwise reinforced by using variational multiscale approach without modifying the kinematic model. The originality of the approach presented herein is in the interpretation of the MPCs application as the solution in a superfluously extended space because of the weakening in the kinematic constraints. It is shown that the perfect bond between the composite beam layers can be recovered by excluding the identified fine-scale effect from the solution of the multiple point constraint application. The convergence characteristic of the finite element formulation is also improved by using the variational multi-scale approach. It is also shown that the fine-scale effects can be represented by using extra fictitious elements and springs, which offers a direct correction technique in modelling of composite beams that is especially useful when access to the numerical procedure is limited.

Using multiple point constraints in finite element analysis of two dimensional contact problems

Two-dimensional elastic contact problems, including normal, tangential, and rolling contacts, are treated with the finite element method in this study. Stress boundary conditions and kinematic conditions are transformed into multiple point constraints for nodal displacements in the finite element method. Upon imposing these constraints into the finite element system equations, the calculated nodal stresses and nodal displacements satisfy stress and displacement contact conditions exactly. Frictional and frictionless contacts between elastically identical as well as elastically dissimilar materials are treated in this study. The contact lengths, sizes of slip and stick regions, the normal and the shear stresses can be found.

Development of a finite-element analysis methodology for the propagation of delaminations in composite structures

Analysing the collapse of skin-stiffened structures requires capturing the critical phenomenon of skin-stiffener separation, which can be considered analogous to interlaminar cracking. This paper presents the development of a numerical approach for simulating the propagation of interlaminar cracks in composite structures. A degradation methodology was introduced in MSC.Marc, which involved the modelling of a structure with shell layers connected by user-defined multiple-point constraints (MPCs). User subroutines were written that employ the virtual crack closure technique (VCCT) to determine the onset of crack growth and modify the properties of the user-defined MPCs to simulate crack propagation. Methodologies for the release of failing MPCs are presented and are discussed with reference to the VCCT assumption of self-similar crack growth. The numerical results obtained by using the release methodologies are then compared with experimental data for a double-cantilever beam specimen. Based on this comparison, recommendations for the future development of the degradation model are made, especially with reference to developing an approach for the collapse analysis of fuselage-representative structures.

Finite Element Analysis of Interface Cracks Using Multiple Point Constraints

A finite element technique to analyse closed-mode interface cracks is proposed in this study. Stress boundary conditions for a closed-tip model are transformed into multiple point constraints (MPCs) for nodal displacements. These constraints are imposed upon the finite element solutions to simulate closed-tip stress fields in crack-tip elements. This technique can deal with small as well as large crack-tip contact zones, and no special elements other than the standard quarter-point elements are needed. Since stresses approach infinity at the crack tip in a quarter-point element, dominant terms are used in deriving MPCs for nodal displacements. Fracture parameters are obtained by using the virtual crack extension method, and numerical results are in good agreement with analytical results.

A new automatic hexahedral mesher based on cutting

AbstractA new method is presented leading to a 20‐node brick element mesh for an arbitrary structure based on a triangulation of the structure's surface. Starting from a hexahedral master mesh encompassing the structure, the elements which are intersected by the triangulation are determined, cut and remeshed. The resulting hexahedral mesh can be tied to the underlying elements by multiple point constraints or can be continued throughout the complete structure leading to a pure 20‐node brick mesh. Several examples show the effectiveness of the method. Copyright © 2001 John Wiley & Sons, Ltd.

Unstructured 20-node brick element meshing

A new method is presented to generate an unstructured 20-node brick element mesh for arbitrary structures. Based on a triangulation of the structure's surface, a 20-node master mesh is generated encompassing the structure. The elements intersected by the triangulation are determined, catalogued, cut and remeshed according to their cutting topology. The new elements external to the structure are discarded, while the others are tied to the uncut subsurface mesh by means of multiple point constraints (MPCs). Alternatively, the remeshing procedure can be continued into the subsurface element layers to obtain a pure 20-node brick element mesh, without any MPCs. Additional smoothing further improves the mesh quality.

Dynamic response of a beam on multiple supports with a moving mass

The dynamic behavior of an Euler beam with multiple point constraints traversed by a moving concentrated mass, a "moving-force moving-mass" problem, is analyzed and compared with the corresponding simplified "moving-force" problem. The equation of motion in matrix form is formulated using Lagrangian approach and the assumed mode method. The effects of the presence of intermediate point constraints in reducing the fluctuation of the contact force between the mass and the beam and the possible separation of the mass from the beam are investigated. The equation of motion and the numerical results are expressed in dimensionless form. The numerical results presented are therefore applicable for a large combination of system parameters.