Non-penetration Condition Research Articles

Liquid Induced Vibrations of Truncated Elastic Conical Shells with Elastic and Rigid Bottoms

A method of estimating natural modes and frequencies of vibrations for elastic shells of revolution conveying a liquid is proposed. The vibration modes of the liquid-filled elastic shells are presented as linear combinations of their own vibration modes without liquid. The explicit expression for fluid pressure is defined using Bernoulli’s integral and potential theory suppositions. Non-penetration, kinematic, and dynamic boundary conditions are applied at the shell walls and on a free liquid surface, respectively. The solution of the hydro-elasticity problem is found out using an effective technique based on coupled finite and boundary element methods. Computational vibration analysis of elastic truncated conical shells with different fixation conditions is accomplished. Sloshing and elastic walls frequencies and modes of liquid-filled truncated conical tanks are estimated. Both rigid and elastic bottoms of shells are considered. Some examples of numerical estimations are provided to testify the efficiency of the developed method

A Contact Problem for Two Plates of the Same Shape Glued along One Edge of a Crack

Under study is the equilibrium problem for two plates with possible contact between them. It is assumed that the plates of the same shape and size are located in parallel without a gap. The clamped edge condition is stated on their lateral boundaries. The deflections of the plates satisfy the nonpenetration condition. There is a vertical crack in the lower layer. Along one edge of the crack, the plates are rigidly glued with each other. The three cases are studied in the paper: In the first case, the both layers are elastic, whereas in the second and third cases, the lower or upper layer respectively is rigid. To describe the displacement of the points of elastic plates, the Kirchhoff–Love model is used. Variational and differential formulations of the problems are derived and the unique solvability of the problems is established.

Two-dimensional turning of thermal flux from normal to lateral propagation in thin metal film irradiated by femtosecond laser pulse

There are various geometrical variants of laser illumination and target design. Important direction of investigations is connected with tightly focused action (spot size may be less than micron) onto a thin metal film: thickness of a film is just few skin-layer depths. Duration of a pulse is τL ∼ 0.1 ps. In these conditions energy absorbed in a skin layer first propagates normally to a surface: gradient ∂Te/∂x dominates, here and below x and y are normal and lateral directions. This process in 1–2 ps homogenizes electron temperature Te along thickness of a film. We consider conditions when a film or is supported by weakly conducting substrate, or is free standing. Therefore all absorbed energy is confined inside the film. At the next stage the internal energy begin to flow along the lateral direction—thus direction of energy expansion is changed from x to y because of the heat non-penetrating boundary condition imposed on the rear-side of the film. At the short two-temperature stage of lateral expansion the thermal conductivity κ is high. After that electron and ion temperatures equilibrates and later on the heat propagates with usual value of κ. Lateral expansion cools down the hot spot on long time scales and finally the molten spot recrystallizes. Two-dimensional approach allows us to consider all these stages from propagation in x direction (normal to a film) to propagation in y direction (along a film).

An enhanced SDOF model to predict the behaviour of a steel column impacted by a rigid body

The transient dynamic response of a steel beam-column subjected to impact loading is a complex phenomenon involving large localized plastic deformations and non-smooth contact interactions. Exposed to high intensity of the contact force generated from impact, the beam-column may undergo large displacement and inelastic deformation. Previous research has shown that a calibrated elasto-plastic single degree of freedom system is able to reproduce both the displacement and the force time-history of a steel beam subjected to non-impulsive loading or low-velocity impact. In these models, the static force-displacement curve is derived from either experiments or detailed 3D nonlinear analysis. Tri-linear resistance function has been extensively used to reproduce the different stages of the response including catenary effects. A rigorous treatment of such a complex problem calls for the use of non-smooth analysis tools to handle the impulsive nature of the impact force, the unilateral constraint, the impenetrability condition and the discontinuity of the velocity in a rigorous manner. In this paper, we present a non-smooth elasto-plastic single degree of freedom model under impact loading that permits the use of arbitrary resistance function. Adopting the non-smooth framework offers tools such as differential measures and convex analysis concepts to deal with unilateral contact incorporating Newton’s impact law. The mid-point scheme is adopted to avoid numerical unrealistic energy decay or blowup. Furthermore, the non-penetration condition is numerically satisfied by imposing the constraint at only the velocity level to guarantee energy-momentum conservation [1]. The explicit expression of resistance functions of the beam that are used in the SDOF model are obtained from a simplified nonlinear static analysis of two beam-column models. In the analysis, a linear relation between normal force and bending moment is assumed for the plastification of the hinges. Two proposals to simplify the explicit expressions of the model’s response behavior are given. Performing an energy-based analysis, we predict maximum displacement that is needed to absorb the kinetic energy arising from the impact for different coefficient of restitution. The numerical examples underline the validity of the model by showing good agreement with the predictions of reference models.

Hydroelastic analysis of underwater rotating elastic marine propellers by using a coupled BEM-FEM algorithm

This paper focuses on the development of a numerical method for analyzing the added mass and damping of a rotating elastic marine propeller. Three-dimensional panel methods in frequency domain combined with the finite element method were employed to study the strongly coupled fluid-structure interactions of the propeller. In order to overcome the computational efficiency problem due to the asymmetric added matrices of the fluid, a mode superposition method in conjunction with Wilson-θ method was employed for calculating the structural responses of the propeller. The validity of the proposed numerical method was confirmed by comparing the present results with experimental data available in the literature and those numerical solutions computed using commercial packages ANSYS and Virtual.Lab Acoustics. The effects of the excitation frequency, inflow velocity, material parameter of propeller and the advance ratio on the added mass and damping of rotating elastic propellers were examined. The results showed that stationary flow may be sufficient for analyzing the wet modes of the propeller at a relatively high excitation frequency, and the non-penetration boundary conditions should be imposed on the deformed blade surface rather than the undeformed surface in the case of relatively lower-frequency excitations. In addition, if the inflow velocity is relatively large, the added damping due to the fluid can significantly affect the unsteady performance of the propeller.

Smoothed profile-lattice Boltzmann method for non-penetration and wetting boundary conditions in two and three dimensions

In this study, the smoothed profile-lattice Boltzmann method (SP-LBM) is proposed to determine the contact line dynamics on a hydrophobic or a hydrophilic curved wall. Two types of smoothed indicator functions are introduced, namely a function that identifies the solid domain for non-slip and non-penetration conditions and a function that denotes the boundary layer for no mass-flux and the wetting boundary conditions. In order to prevent fluid penetration into the solid boundary, the fluid-solid interaction force is computed based on the definition of the fluid velocity as proposed by Guo et al. [1]. In order to implement the Neumann boundary conditions for the order parameter and the chemical potential, the fluxes from the solid surfaces are distributed to relevant physical valuables through a smoothed profile. Several two-dimensional and three-dimensional numerical investigations including those determining the Couette flows, flow around a circular cylinder, transition layer on a wetting boundary, and dynamic behavior of a droplet on a flat or curved plate demonstrate the efficiency of the present method in calculating the contact angle of a droplet on curved surfaces with wall impermeability. The present model provides a simple algorithm to compute the surface normal vector and contact line dynamics on an arbitrarily shaped boundary by using a smoothed-profile.

On the Efficient Reconstruction of Displacements in FETI Methods for Contact Problems

The final step in the solution of contact problems of elasticity by FETI-based domain decomposition methods is the reconstruction of displacements corresponding to the Lagrange multipliers for ''gluing'' of subdomains and non-penetration conditions. The rigid body component of the displacements is usually obtained by means of a well known but quite complex formula, the application of which requires reassembling and factorization of some large matrices. Here we propose a simple formula which is applicable to many variants of the FETI based algorithms for contact problems. The method takes a negligible time and avoids reassembling or factorization of any matrices.

Vibrations of a beam between two rigid stops: strong solutions in the framework of vector-valued measures

We study a problem that models the dynamics of an elastic beam vibrating between two rigid stops, so we use the Signorini non-penetration condition to describe the contact process. This allows for impacts and velocity jumps. Motivated by the need to better understand this kind of dynamics, we introduce a new formulation of the problem in the framework of vector-valued measures, somewhat similar to the case of a discrete mechanical system. We prove the existence of a strong solution and establish the main properties of the reaction shear stress that acts on the system at impacts, which is a measure with a singular part.

A fast and automatic full-potential finite volume solver on Cartesian grids for unconventional configurations

To meet the requirements of fast and automatic computation of subsonic and transonic aerodynamics in aircraft conceptual design, a novel finite volume solver for full potential flows on adaptive Cartesian grids is developed in this paper. Cartesian grids with geometric adaptation are firstly generated automatically with boundary cells processed by cell-cutting and cell-merging algorithms. The nonlinear full potential equation is discretized by a finite volume scheme on these Cartesian grids and iteratively solved in an implicit fashion with a generalized minimum residual (GMRES) algorithm. During computation, solution-based mesh adaptation is also applied so as to capture flow features more accurately. An improved ghost-cell method is proposed to implement the non-penetration wall boundary condition where the velocity-potential of a ghost cell is modified by an analytic method instead. According to the characteristics of the Cartesian grids, the Kutta condition is applied by specially computing the gradients on Kutta-faces without directly assigning the potential jump to cells adjacent wake faces, which can significantly improve the solution converging speed. The feasibility and accuracy of the proposed method are validated by several typical cases of sub/transonic flows around an ONERA M6 wing, a DLR-F4 wing-body, and an unconventional figuration of a blended wing body (BWB). The validation cases demonstrate a fast convergence with fully automatic grid treatment and computation, and the results suggest its capacity in application for aircraft conceptual design.

Singular invariant integrals for elastic bodies with thin elastic inclusions and cracks

Equilibrium problems for an elastic body with partially delaminated thin elastic inclusions are considered. The inclusions are modeled within the framework of the Euler–Bernoulli and Timoshenko models of elastic beams. The presence of delamination means the existence of a crack between the inclusion and the elastic matrix. Displacements of the opposite crack faces are constrained with nonpenetration conditions. Formulas of the Griffith type giving the first derivatives of energy functionals with respect to the crack length are established. It is shown that the formulas for the derivatives admit representation in the form of invariant integrals independent of the smooth closed curve surrounding the crack tip. The obtained invariant integrals consist of the sum of regular and singular parts and are analogues of the classical Eshelby–Cherepanov–Rice J-integral.

Domain decomposition method for crack problems with nonpenetration condition

The work deals with an iteration method for numerical solving the equilibrium problem of two-dimensional elastic body with a crack under the nonpenetration condition. The method is based on the domain decomposition and Uzawa’s algorithm. To construct an algorithm, the domain is partitioned into two subdomains whose common boundary contains the crack. In each subdomain the linear problems are solved. We use Lagrangian multipliers to couple the solutions and provide the nonpenetration condition on the crack.

A weighted point-based formulation for isogeometric contact

In the context of isogeometric contact analysis, surfaces of objects can be described smoothly due to the high continuity of the involved shape functions. This facilitates construction of a continuous normal field which may be beneficial for modeling of contact problems. This property allows for stable collocation of the virtual contact work using considerable less evaluation points than numerical integration, while the approximation quality of the computational results is almost the same. In this context, a weighted point-based method (PTS+) is introduced as an extension of the Point-To-Segment method (PTS) proposed previously by Matzen et al. (2013). The weights enable transmission of contact stresses instead of contact forces along the contact surfaces. In normal direction the non-penetration condition is enforced by the Lagrange multiplier method, combined with the penalty method in tangential direction. Corresponding to the collocation weights, a constant or NURBS Lagrange multiplier field is introduced. A comparison between the collocation-based contact formulations PTS, PTS+ and a mortar method is drawn, showing that weighted collocation is able to achieve almost the same accuracy as the integration-based mortar formulation.

Shape optimization of rigid inclusions for elastic plates with cracks

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff–Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem.

The Griffith formula and J‐integral for elastic bodies with Timoshenko inclusions

The paper contains an analysis of a two‐dimensional equilibrium problem for an elastic body with a thin elastic inclusion. The thin elastic inclusion is modeled within the framework of Timoshenko beam theory. There is a crack on the interface between two media, displacements of the opposite crack faces are constrained with nonpenetration conditions. We derive the Griffith formula, which gives the first derivative of the energy functional with respect to the crack length. It is proved that the formula for the derivative can be represented as a path‐independent integral along a smooth curve surrounding the crack tip. The invariant integral consists of a regular part and a singular part and is an analogue of the classical Eshelby–Cherepanov–Rice J‐integral.

A vortex approach for unsteady insect flight analysis in 2D

This paper considers 2D insect wing motion in which the flow field does not change in the out-of-plane direction. When modelling complex phenomena, simpler, but not overly simple, analysis techniques become imperative. We achieve the balance of simplicity and accuracy by a technique based on the vortex. The wing is represented by its chord, described by a line in 2D. During the unsteady flapping, a time-dependent circulation pattern is developed over the wing, modelled by the distribution of discrete line vortices. The magnitudes of the bound vortices on the wing are determined by the non-penetration condition. Two vortices at the wing’s leading and trailing edges are shed. The velocity is determined by the bound and wake vortices using 2D Biot–Savart law, which is also used to convect wake vortices. The entire cycle is repeated as the time progresses. While establishing a simple affordable numerical method for flapping wing analysis and highlighting its remarkable performance, the limitation of the metho...

An entropy-based evaluation of contact forces in continuum mechanics of elastic structures

The maximum entropy principle has proved to be a versatile tool for solving problems in many fields. In this paper, we extend entropy and its use in statistical physics to evaluate contact forces in the continuum mechanics of elastic structures. Each potential contact node of an elastic structure discretized by finite elements along with the normalized contact force on the node is considered as a system and all potential contact nodes together with their normalized contact forces are considered as a canonical ensemble, with the normalized contact force of each node representing the microstate of the node. The product of non-penetration conditions for potential contact nodes and the normalized nodal contact forces then act as an expectation that its value will be zero, and maximizing the entropy under the constraints of the expectation and the minimum potential energy principle results in an explicit probability distribution for the normalized contact forces that shows the relation between contact forces and displacements in a formulation similar to the formulation for particles occupying microstates in statistical physics. Moreover, an iterative procedure that solves a series of isolated systems to find the contact forces is presented, with a novel termination condition. Finally, some examples are examined to verify the correctness and efficiency of the procedure.

Node-to-node scheme for three-dimensional contact mechanics using polyhedral type variable-node elements

A node-to-node contact scheme, which is applicable for three-dimensional contact analyses involving large deformations, is developed with the aid of polyhedral elements. The key issue is to transform nonmatching meshes into matching meshes in a seamless manner. Because polyhedral elements are allowed to have arbitrary numbers of polygonal faces and nodes, they can be used as transition elements for coupling nonmatching meshes. In this paper, the polyhedral elements make it possible to always maintain node-to-node contact during the contact deformation. The present approach guarantees that the patch test is passed and the nonpenetration condition is satisfied, and hence it yields smoother contact pressure with faster convergence than the conventional node-to-surface or surface-to-surface contact scheme.

Evaluation of mechanical properties using spherical ball indentation and coupled finite element–element-free galerkin approach

ABSTRACTIn the present work, the mechanical properties of pressure tube material (Zr-Nb2.5) are evaluated using the coupled finite element–element-free Galerkin approach. Penalty approach is used to impose contact constraints and non-penetration condition at the interface. An efficient node-to-segment algorithm is employed to model the contact behavior. An updated Lagrangian approach is used to model the large deformation. Loading and unloading response of the indentation process is analyzed using von-Mises and Gurson-Tvergaard-Needleman (GTN) plasticity models. In multiple indentations, the indentation depth is progressively increased up to a maximum specified limit with partial unloading. Load-indentation depth curves are used to extract the flow properties of the material.

Existence of an optimal size of a rigid inclusion for an equilibrium problem of a Timoshenko plate with Signorini-type boundary condition

We study the contact problems for elastic plates with a rigid inclusion. We consider the case of frictionless contact between the rigid part of the plate and a rigid obstacle. The contact is modeled with the Signorini-type nonpenetration condition. The deformation of the transversely isotropic elastic part of the plate is described by the Timoshenko model. We analyze the dependence of solutions to the contact problems on the size of rigid inclusion. The existence of a solution to the optimal control problem is proved. For that problem, the cost functional characterizes the deviation of the displacement vector from a given function, whereas the size parameter of rigid inclusion is chosen as the control function.

Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle

The equilibrium problems for homogeneous and inhomogeneous plates with a crack is considered. We impose the nonpenetration condition, which has the form of inequality (such as Signorini type condition), on the crack faces. In this paper, we deal with the cases that the crack intersects the external boundary at zero angle (on the mid-plane). Using the fictitious domain method, we establish the unique solvability of four equilibrium problems for different cases of non-Lipschitz domains. In these cases a precise connection between equilibrium problems for a plate contacting with a rigid obstacle and for a plate with a crack is identified.