Solution Of Contact Problem Research Articles

Geodesic algorithm: new approach to optimization of temporary fastener arrangement in airframe assembly process

PurposeThe purpose of the current study is development of effective and fast algorithm for optimization the arrangement of temporary fasteners during aircraft assembly.Design/methodology/approachCombinatorial nature, uncertain input data, sensitivity to mechanical properties and geometric tolerances are the specific features of the fastening optimization problem. These characteristics make the problem-solving by standard methods very resource-intensive because the calculation of the objective function requires multiple solution of contact problems. The work provides an extended description of the geodesic algorithm (GA) which is a novel non-iterative optimization approach avoiding multiple objective function calculations.FindingsThe GA makes it possible to optimize the arrangement of temporary fasteners during the different stages of the assembly process. The objective functions for the optimization are number of installed fasteners and quality of contact between joined parts. The mentioned properties of the GA also make it possible to introduce an automatic procedure for optimizing fastener arrangement into everyday practice of aircraft manufacturing.Practical implicationsThe algorithm has been applied to optimization of the assembly process in Airbus company.Originality/valuePerformance of the GA is orders of magnitude greater than standard optimization algorithms while maintaining the quality of results. The use of the assembly process specifics is the main limitation of the GA, because it cannot be automatically applied to optimization problems in other areas. High speed of work and quality of the results make it possible to use it for real optimization problems on assembly line in the production of commercial airliners.

Use of patellofemoral digital twins for patellar tracking and treatment prediction: comparison of 3D models and contact detection algorithms.

Introduction: Poor patellar tracking can result in painful contact pressures, patella subluxation, or dislocation. The use of musculoskeletal models and simulations in orthopedic surgeries allows for objective predictions of post-treatment function, empowering clinicians to explore diverse treatment options for patients. Although a promising approach for managing knee surgeries, the high computational cost of the Finite Element Method hampers its clinical usability. In anticipation of minimal elastic deformations in the involved bodies, the exploration of the Multibody Dynamics approach emerged as a viable solution, providing a computationally efficient methodology to address clinical concerns related to the knee joint. Methods: This work, with a focus on high-performance computing, achieved the simulation of the patellofemoral joint through rigid-body multibody dynamics formulations. A comparison was made between two collision detection algorithms employed in the simulation of contact between the patellar and femoral implants: a generic mesh-to-mesh collision detection algorithm, which identifies potential collisions between bodies by checking for proximity or overlap between their discretized mesh surface elements, and an analytical contact algorithm, which uses a mathematical model to provide closed-form solutions for specific contact problems, but cannot handle arbitrary geometries. In addition, different digital twins (3D model geometries) of the femoral implant were compared. Results: Computational efficiency was considered, and histories of position, orientation, and contact force of the patella during the motion were compared with experimental measurements obtained from a sensorized 3D-printed test bench under pathological and treatment scenarios. The best results were achieved through a purely analytical contact detection algorithm, allowing for clinical usability and optimization of clinical outcomes.

Solutions for contact problems applied to bevel gears

The article presents solutions to flat contact problems applied to straight-tooth bevel gears, considering both linear and nonlinear dependencies between elastic displacements of the teeth and the resulting stresses. Maximum contact stresses and contact patch dimensions are determined, accounting for both linear and nonlinear relationships between elastic deformations of the teeth and stresses. These solutions are based on a novel contact strength theory. It is demonstrated that the load capacity of bevel gears, considering contact stresses and nonlinear dependencies between elastic deformations and stresses, increases by 1.113 times.

Mortar Method for 2D Elastic Bounded Contact Problems

Abstract This paper presents a contribution to the field of numerical solutions for contact problems, which pose significant challenges in engineering and simulations. Specifically, we address the intricate task of connecting bodies that have been discretized using non-conforming and non-overlapping meshes. Our primary focus lies in investigating the efficacy of the mortar method with a segment-to-segment approach. In this context, we provide a concise overview of the underlying theoretical framework and present our implementation in the MATLAB programming environment. To ascertain the reliability and accuracy of our proposed methodology, we conduct a rigorous validation study by comparing the outcomes obtained from our implementation with those derived from the widely adopted commercial software, ANSYS. To enable a comprehensive evaluation, we select specific benchmark problems that involve the interaction of two elastic bodies. Through a meticulous analysis and comparison of results, we demonstrate the effectiveness and robustness of our approach. The findings of this study contribute substantively to the advancement of numerical techniques for solving contact problems. The validated methodology not only establishes a solid foundation for future research endeavors but also offers a reliable framework for conducting simulations in this domain. Furthermore, the insights gained from this study can potentially facilitate the development of more efficient and accurate computational algorithms for addressing contact problems encountered in various engineering applications.

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.

Numerical strategy for solving general C1‐continuous beam‐to‐beam contact problems

AbstractIn the context of a multibody numerical environment, when aiming at representing the contact interaction between bodies one needs the inclusion of special mathematical formulations. They can be geometrically described as surface‐to‐surface or line‐to‐line contacts, for instance. The last one is usually the natural choice when addressing beam‐to‐beam contact. In the present work, a geometrically nonlinear structural formulation is combined with a beam‐to‐beam spline‐based contact formulation with a particular numerical strategy to address challenging contact problems. The numerical strategy consists of introducing criteria to evaluate, characterize and decide about the contact problem solutions. According to these criteria, a contact problem solution may be accepted, rejected, or proceed to a degeneration process (as a simplification of the original tentative for the geometric description). The main advantages of the proposed technique are the lack of special contact formulations for ill‐defined contact problems and the automatic switch between degenerated and nondegenerated scenarios. Therefore, one can naturally handle scenarios of pointwise or conformal contact, even in situations of transition between such categories along the model evolution. The adopted spline‐based formulation includes normal and tangential contact contributions. To test the proposed strategy, several pointwise and conformal contact examples are proposed.

ON CONTACT PROBLEMS FOR TWO STAMPS AND A NEW TYPE OF CRACK MODEL

In this work, for the first time, an exact solution of the contact problem of interaction with a multilayer base of two semi-infinite stamps, the ends of which are parallel to each other, is constructed. Stamps are assumed to be absolutely rigid, and the distance between them can have any finite value. The task is an important stage in the algorithm for constructing models of a new type of crack in materials of different rheologies. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been poorly studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The problem considered in the article can be considered as the result of the formation of a new type of crack with absolutely rigid banks and deformable lower boundary. Thanks to it, after the solution, it becomes possible to switch to deformable stamps and a crack of a new type in the rheological medium. The solution of this problem turned out to be possible due to the construction of exact solutions of the Wiener–Hopf integral equations on a finite segment. The paper shows how the solution of one of the previously unsolved problems allows us to investigate and solve exactly other problems, to identify previously unknown properties and resonances. As a result of constructing an exact solution to the problem, the fact that the solution of dynamic contact problems for stamp systems is not unique was confirmed and a dispersion equation for finding resonant frequencies was constructed.

Analytical solution for contact and crack problem ın homogeneous half-plane

Analytical solution for contact and crack problem ın homogeneous half-plane

Multigrid method for the solution of thermal elastohydrodynamic lubrication point contact problem with surface asperities

The paper presents, the numerical investigation of point contact thermal elasto-hydrodynamic lubrication (EHL) with surface asperities are analyzed. The effect of temperature and surface roughness on fluid film thickness is studied in detail. The governing equations comprises Reynolds, film thickness, load balance and energy equations with appropriate boundary conditions. The second order finite difference approximation is used to discretize the governing equations and the resultant nonlinear system of algebraic equations is solved using Multigrid V-cycle with full approximation scheme (FAS) technique. Multi level multi integration (MLMI) technique is employed to solve the film thickness equation. The obtained results are illustrated in the form of graphs and tables which are comparable with earlier findings. The film thickness profiles shows dimple near to the outlet region due to temperature-viscosity wedge mechanism. Isothermal minimum film thickness is higher than the thermal minimum film thickness. Minimum film thickness is much smaller due to slide to roll ratio is positive ascompared to negative, whereas the behavior of central film thickness is contrast as that of minimum film thickness.

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

Equilibrium finite element and error estimation for frictional contact problem based on linear complementarity problem formulation

AbstractThis article proposes a novel equilibrium finite element method (EFEM) for equilibrated solution of frictional contact problem. The development of such EFEM is mainly two‐fold. On the one hand, the traction‐based equilibrium element is employed to construct the discrete equilibrated stress field. This equilibrium element, using edge tractions as basic variables, is found straightforward to deal with the friction and contact constraints on the contact surface. On the other hand, the Coulomb frictional contact problem is recast into a linear complementarity problem (LCP), with which the algebraic complementarity formulation can be obtained with the help of the constructed equilibrated stress field. A remarkable property of the proposed EFEM is that the equilibrated solution, in conjunction with the compatible solution of the traditional FEM, can provide reliable error estimation for frictional contact within the framework of dual analysis. Numerical examples are finally conducted to see the performance and the ability to reliable error estimation of the proposed EFEM.

On contact problems with a deformable punch and variable rheology

The paper presents for the first time one of the methods for studying and solving contact problems with a deformed stamp for those cases when there is a need to change the rheology of the stamp material. It is based on a new universal modeling method previously published by the authors, which is used in boundary value problems for systems of partial differential equations. With its help, solutions of complex vector boundary value problems for systems of differential equations can be decomposed into solutions of scalar boundary value problems for individual differential equations. Among them, the Helmholtz equations are the simplest. Solutions to scalar boundary value problems are represented as fractals, self-similar mathematical objects first introduced by the American mathematician B. Mandelbrot. The role of fractals is performed by packed block elements. The transition from systems of differential equations in partial derivatives to individual equations is carried out using the transformation of academician B. G. Galerkin or representation by potentials. It is known that the solutions of dynamic contact problems with a deformable stamp of complex rheology are cumbersome, and their study is always difficult. The problem is complicated by the presence of discrete resonant frequencies in such problems, which were once discovered by Academician I. I. Vorovich.

EXACT SOLUTION OF CONTACT PROBLEMS IN A FINITE-WIDTH BAND ON A MULTILAYER MEDIUM

In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.

Solution of contact problems of anisotropic plates bending on an elastic base using the compensating loads method

In critical structures used elements consisting of anisotropic materials with concentrated force or moment applied to them are often found. In order to equalize the internal force factors, as well as to ensure the required strength and rigidity of the structure, in places where a local load is applied, it is necessary to strengthen the plate with a rigid lining (stamp). There are several ways to solve contact problems. The way seems simpler if it is possible to determine the exact fundamental solution (FS) for the plate. The exact FS significantly reduces the amount of computational work due to the fact that the boundary conditions and the conditions of conjugation of solutions at the boundary of the contact area are met in advance. It remains only to formulate the conditions for the compatibility of movements in the contact area. With this formulation, the integral equation of the contact condition becomes a singular integral equation of the first kind. Such equations are reduced by regularization to Fredholm equations of the second kind and the problem becomes mathematically and physically correct. The paper considers contact problems without taking into account tangential interaction between objects. For such problems, the following way of simplification is possible: replacing the singular core for a half-space with a core for a layer of constant thickness modeled by an elastic base with a single base coefficient. Such a replacement for thin shells is quite justified and gives solutions that are in good agreement with the solutions of the theory of elasticity. The deflection of the plate is sought using the compensating loads method. To work out the methodology, a test contact problem was solved.

An accelerated method of Uzawa algorithm in contact problems

A lot of methods are applied in the numerical solution of contact problems, among which the most widely used method is known as Uzawa algorithm. A contact problem can transform into a saddle problem. We use Uzawa algorithm for this saddle format. Because this form can get both displacements for nodes and outward normal components of stress vectors for boundary nodes. Although the Uzawa algorithm is easy to implement, its convergent velocity is slow. Taking this algorithm as a fixed point iteration, we can use the Anderson acceleration to improve the convergent velocity.

Frictional Energy Dissipation in Partial Slip Contacts of Axisymmetric Power-Law Graded Elastic Solids under Oscillating Tangential Loads: Effect of the Geometry and the In-Depth Grading

Due to the rapid development of additive manufacturing, a growing number of components in mechanical engineering are made of functionally graded materials. Compared to conventional materials, they exhibit improved properties in terms of strength, thermal, wear or corrosion resistance. However, because of the varying material properties, especially the type of in-depth grading of Young’s modulus, the solution of contact problems including the frequently encountered tangential fretting becomes significantly more difficult. The present work is intended to contribute to this context. The partial-slip contact of axisymmetric, power-law graded elastic solids under classical loading by a constant normal force and an oscillating tangential force is investigated both numerically and analytically. For this purpose, a fictitious equivalent contact model in the mathematical space of the Abel transform is used since it simplifies the solution procedure considerably without being an approximation. For different axisymmetric shaped solids and various elastic inhomogeneities (types of in-depth grading), the hysteresis loops are numerically generated and the corresponding dissipated frictional energies per cycle are determined. Moreover, a closed-form analytical solution for the dissipated energy is derived, which is applicable for a breadth class of axisymmetric shapes and elastic inhomogeneities. The famous solution of Mindlin et al. emerges as a special case.

On the issue of a new solution of the materials resistance contact problem on compression of elastic cylinders in contact with parallel generators

As a result of logarithmic singularity in the plane classical problem of elastically deformable material mechanics, it is proved that the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic, and physically linear material is not applicable. In the special case of elastic interaction of a cylinder with a half-plane, it is established that the convergence becomes equal to infinity. This paradoxical result confirms the nadequacy of Flaman’s model of a simple radial stress state in determining displacements. Based on this model, it is possible to determine only the stresses in parallel contacting cylinders, while the calculation of displacements, in this case, is not possible. Based on apreviously developed and mathematically approximated by the authors flat design scheme of Flaman the algorithm exception of conflicts has been proposed. The algorithm is based on the integral Fredholm equation solution and can be seen as a new fundamental and applied elasticity theory problem, which is of great importance when assessing the contact of refined strength and stiffness of the cylindrical parts of the supporting structures subject to the general and local deformations (cylindrical rollers, gears, pavements with their seal steel rollers, etc.).

Wear rate and biocompatibility of pre and post UV irradiated UHMWPE for tribo-pair in total knee replacement application

Ultra-high molecular weight polyethylene (UHMWPE) and CoCr alloy are popular tribo-pair in total knee replacement. Wear in the liner is a major failure reason for knee implant. Therefore, this work focuses on an approach for reducing the wear rate by irradiating the UHMWPE specimens using Ultraviolet (UV) radiation. The powder of UHMWPE was molded into a plate by microwave-assisted compression (MAC) molding. The UV radiations of intensity 0.025 J/cm2 were irradiated on the MAC molded UHMWPE specimens. The wear rate was determined using a pin on the disc wear tribometer using the pre and post-irradiated UHMWPE specimens as a pin and CoCr alloy as the disc. The pre and post-irradiated UHMWPE sliding was done at the load of 40 N for 1500 m under dry conditions. The reduction in the wear rate recorded was 56% after UV irradiation. The surface morphology of the worn specimens was done using scanning electron microscopy (SEM) and the 3D surface mapping technique. The obtained results of wear rate were validated numerically by implementing the contact problem solution in Archard’s wear law using user-subroutine on Python. The experimental and numerically obtained results were in good agreement. The biological response of pre and post-irradiated specimens was evaluated by hemolysis assay, cellular compatibility against peripheral blood mononuclear cells, platelet adhesion, and in vitro degradation under a simulated blood fluid environment.

Regularization for a class of quasi-variational-hemivariational inequalities

The aim of the present paper is to study the solvability and regularization for a class of multivalued quasi-variational–hemivariational inequalities in reflexive Banach spaces. By applying the Kluge fixed point theorem and the Minty technique, we prove the solvability of the considered multivalued quasi-variational–hemivariational inequality, based on which some convergence results are obtained by introducing its regularization problem with the help of regularization operator. The applicability of the obtained abstract results is established by a mathematical model of a frictional contact problem with a class of elastic material, where the existence and stability results for the weak solution of contact problem are studied.

An innovative and efficient solution for axisymmetric contact problem between structure and half-space

In this paper, with displacement increment as variables, the scaled boundary finite element equations of the axisymmetric plate and half-space are respectively derived based on the principle of virtual work. The dual vector is introduced to solve the boundary stiffness matrix. Linear complementary contact conditions are formulated as B-differentiable equations. With discrete, static assumptions of small displacement, B-differentiable equations system was formed by combining the contact conditions and the stiffness matrix of the plate-foundation. The equations can be iteratively solved by B-differentiable Newton method. In numerical experiments, the contact stress, slip and contact status of each contact node pair are compared with the results of the augmented Lagrangian contact algorithm based on FEM. When adopting different load types, thickness-to-radius ratio, plate-foundation stiffness ratio and friction coefficient, the axisymmetric contact problems between plate and half-space are discretized by 120 one-dimensional first-order elements. The applicability, accuracy and effectiveness of B-SBFEM for axisymmetric contact problems are verified by numerical examples.