Rigid Bodies Of Arbitrary Shape Research Articles

Motion of Rigid Bodies of Arbitrary Shape in a Viscous Incompressible Fluid: Wellposedness and Large Time Behaviour

Motion of Rigid Bodies of Arbitrary Shape in a Viscous Incompressible Fluid: Wellposedness and Large Time Behaviour

Large-Time Behavior of a Rigid Body of Arbitrary Shape in a Viscous Fluid Under the Action of Prescribed Forces and Torques

Let $${\mathcal B}$$ be a sufficiently smooth rigid body (compact set of $${\mathbb R}^3$$ ) of arbitrary shape moving in an unbounded Navier–Stokes liquid under the action of prescribed external force, $$F $$ , and torque, $$M $$ . We show that if the data are suitably regular and small, and $$F $$ and $$M $$ vanish for large times in the $$L^2$$ -sense, there exists at least one global strong solution to the corresponding initial-boundary value problem. Moreover, this solution converges to zero as time approaches infinity. This type of results was known, so far, only when $${\mathcal B}$$ is a ball.

Corrigendum: On the justification of the Foldy–Lax Approximation for the Acoustic Scattering by Small Rigid Bodies of Arbitrary Shapes

We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number M and the diameter a of the obstacles as well as the minimum distance d between them under which the Foldy-Lax approximation is valid. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering.

Forward Statics of Tensegrity Robots With Rigid Bodies Using Homotopy Continuation

Tensegrity robots are composed of simple members (rigid bodies and tensile cables) and have features like lightweight, compliance, and robustness, thus representing a promising alternative to soft and rigid robots. Forward statics, which determines the robot pose in static equilibrium, is very helpful for the design of tensegrity robots. However, existing approaches are not efficient or not general enough for Class-<inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> tensegrity robots, where up to <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> rigid bodies can be connected. This letter proposes a general approach for static modeling of tensegrity robots with rigid bodies of arbitrary shapes, using natural coordinates. Introducing an additional set of variables transforms the static equation into a polynomial system with four kinds of parameters: cable rest lengths, deformations, stiffness coefficients, and external forces. Then, the forward statics problem appears as a path-following problem of a parameter homotopy, which is efficiently solved by the homotopy continuation method. Simulations illustrate how the forward statics can be used to evaluate the achievable range and the strength of a 2D tensegrity manipulator and a 3D tensegrity spine. Finally, we conduct hardware experiments on a tensegrity manipulator prototype, validating the accuracy of the proposed approach.

Adhesive contact between a rigid body of arbitrary shape and a thin elastic coating

Application of the principle of energy balance to a rigid indenter in contact with a thin elastic layer on a flat rigid substrate provides a very simple derivation of the detachment criterion which earlier has been obtained by much more complicated asymptotic analysis. The simple criterion is additionally confirmed by the fully three-dimensional simulations of contact with a coated rigid substrate using the recently developed formulation of the boundary element method for coated media. The found detachment criterion is applied to contact of indenters of various shape. In the case of flat-ended indenters, the adhesive strength occurs to be proportional to the area of the face of the indenter (independently of the shape). The asymptotic criterion is also used for calculation of the adhesion strength of indenters having arbitrary shape and is illustrated with a case study of a contact of a rough indenter with a coated substrate.

Simulating the anchor lifting maneuver of ships using contact detection techniques and continuous contact force models

Designing the geometry of a ship’s hull to guarantee a correct anchor maneuver is not an easy task. The engineer responsible for the design has to make sure that the anchor does not jam up during the lifting process and the position adopted by the anchor on the hull is acceptable when it is completely stowed. Some years ago, the design process was based on wooden scale models of the hull, anchor and chain links, which are expensive, their building process is time consuming and they do not offer the required precision. As a result of this research, a computational tool to simulate the anchor maneuver of generic ships given by CAD models was developed, and it is proving to be very helpful for the naval engineers. In this work, all the theory developed to simulate anchor maneuvers is thoroughly described, taking into account both the behavior of the anchor and the chain. To consider the contact forces between all the elements involved in the maneuver, a general contact algorithm for rigid bodies of arbitrary shapes and a particular contact algorithm for the chain links have been developed. In addition to the contact problem, aspects like the dynamic formulation of the equations of motion or the static equilibrium position problem are also covered in this work. To test the theory, a simulation of the anchor lifting maneuver of a ship is included as a case study. In spite of the motivation to develop the theory, the algorithms derived in this work are general and usable in any other multibody simulation with contacts, under the scope of validity of the models proposed. To illustrate this, the simulation of a valve rocker arm and cam system is accomplished, too. The computational and contact detection algorithms derived have been implemented in MBSLIM, a library for the dynamic simulation of multibody systems, and MBS model, a library for the contact detection and 3D rendering of multibody systems.

Brownian dynamics of confined rigid bodies.

We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.

Reduction of Arbitrary Rigid Bodies to Inertially Equivalent Discrete Systems of Material Points

In a previous paper of the authors, a general method was presented for the reduction of a rigid plane plate to a discrete system of material points, with equivalent inertial properties (mass, center of mass, tensor of inertia). The present paper generalizes the method for rigid bodies of arbitrary shape, i.e. for material volumes, as well as for curved shells. It is shown that a homogenous ellipsoid can be reduced to a system of seven material points placed in significant geometrical points of the body. Next, starting from the concept of ellipsoid of inertia, an equivalent homogenous ellipsoid is determined for an arbitrary body. The method simplifies considerably the calculation of various mechanical quantities, such as moments and products of inertia with respect to rotated Cartesian coordinate systems, angular momentum and kinetic energy, of rigid bodies part of all types of mechanical devices or structures.

Hamiltonian Dynamics of Several Rigid Bodies Interacting with Point Vortices

We derive the dynamics of several rigid bodies of arbitrary shape in a 2-dimensional inviscid and incompressible fluid, whose vorticity field is given by point vortices. We adopt the idea of Vankerschaver et al. (2009) to derive the Hamiltonian formulation via symplectic reduction from a canonical Hamiltonian system. The reduced system is described by a non-canonical symplectic form, which has previously been derived for a single, circular disk using heavy differential-geometric machinery in an infinite-dimensional setting. In contrast, our derivation makes use of the fact that the dynamics of the fluid, and thus the point vortex dynamics, is determined from first principles. Using this knowledge we can directly determine the dynamics on the reduced, finite-dimensional phase space, using only classical mechanics. Furthermore, our approach easily handles several bodies of arbitrary shapes. From the Hamiltonian description we derive a Lagrangian formulation, which enables the system for variational time integrators. We briefly describe how to implement such a numerical scheme and simulate different configurations for validation.

On the Justification of the Foldy--Lax Approximation for the Acoustic Scattering by Small Rigid Bodies of Arbitrary Shapes

We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number $M$ and the diameter $a$ of the obstacles as well as the minimum distance $d$ between them under which the Foldy-Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as it is done sometimes in the literature, then this condition is $(M-1)\frac{a}{d^2} <c$, with an appropriate constant $c$, while if we use double layer potentials then a weaker condition of the form $\sqrt{M-1}\frac{a}{d} <c$ is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters $M, a$ and $d$. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering.

A new approximate analytical solution for arbitrary Stokes flow past rigid bodies

A method of computing general Stokes flows in the presence of rigid boundaries of arbitrary shape is proposed. The solution satisfies the governing field equations exactly and the boundary conditions approximately. The method has been illustrated with three examples. The advantage of the method lies in the ease of implementation for rigid bodies of arbitrary shape, providing an approximate but analytical solution throughout the domain.

A ‘reciprocal’ theorem for the prediction of loads on a body moving in an inhomogeneous flow at arbitrary Reynolds number

AbstractSeveral forms of a theorem providing general expressions for the force and torque acting on a rigid body of arbitrary shape moving in an inhomogeneous incompressible flow at arbitrary Reynolds number are derived. Inhomogeneity arises because of the presence of a wall that partially or entirely bounds the fluid domain and/or a non-uniform carrying flow. This theorem, which stems directly from Navier–Stokes equations and parallels the well-known Lorentz reciprocal theorem extensively employed in low-Reynolds-number hydrodynamics, makes use of auxiliary solenoidal irrotational velocity fields and extends results previously derived by Quartapelle &amp; Napolitano (AIAA J., vol. 21, 1983, pp. 911–913) and Howe (Q. J. Mech. Appl. Maths, vol. 48, 1995, pp. 401–426) in the case of an unbounded flow domain and a fluid at rest at infinity. As the orientation of the auxiliary velocity may be chosen arbitrarily, any component of the force and torque can be evaluated, irrespective of its orientation with respect to the relative velocity between the body and fluid. Three main forms of the theorem are successively derived. The first of these, given in (2.19), is suitable for a body moving in a fluid at rest in the presence of a wall. The most general form (3.6) extends it to the general situation of a body moving in an arbitrary non-uniform flow. Specific attention is then paid to the case of an underlying time-dependent linear flow. Specialized forms of the theorem are provided in this situation for simplified body shapes and flow conditions, in (3.14) and (3.15), making explicit the various couplings between the body’s translation and rotation and the strain rate and vorticity of the carrying flow. The physical meaning of the various contributions to the force and torque and the way in which the present predictions reduce to those provided by available approaches, especially in the inviscid limit, are discussed. Some applications to high-Reynolds-number bubble dynamics, which provide several apparently new predictions, are also presented.

A new algorithm for solving the multi-indentation problem of rigid bodies of arbitrary shapes on a viscoelastic half-space

In this paper the contact problem between rigid indenters of arbitrary shapes and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenters changes and a few contact areas appeared. We wish to find the relations which link the pressure distribution, the resultant forces on the indenters and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shapes the problem may be solved numerically by using the matrix inversion method, extended to viscoelastic cases [1]. But when the problem involves a large number of points the matrix inversion method can become very time-consuming. Here the problem is solved using an alternative scheme, called the two-scale iterative method. In this method the local matrix inversion method is used at the micro-scale for each contact area to compute the pressure distribution taking into account interacting effect (the forces on the other contact areas which can be calculated at the macro-scale) between indenters. Two algorithms were proposed. The first algorithm takes into account the distribution of forces on the other contact areas and the second is the approximation of the first algorithm and takes into account the resultant forces on the other contact areas. The method was implemented for a simple configuration of seven spherical indenters, seven spherical-ended cylindrical indenters and seven flat-ended cylindrical indenters as well as for a more complex configuration of 12 randomly positioned indenters of arbitrary shapes: spherical-ended cylindrical, flat-ended cylindrical, conical and cylindrical indenters (finite cylindrical shape with its curved face). This last case is more difficult as the indenting geometry does not have an axisymmetric profile. For all these cases the two-scale iterative method permits to find the pressure distribution and the contact forces versus the penetration. It can be validated by comparing the numerical results to the numerical results obtained with the matrix inversion method.

Langevin dynamics for rigid bodies of arbitrary shape

We present an algorithm for carrying out Langevin dynamics simulations on complex rigid bodies by incorporating the hydrodynamic resistance tensors for arbitrary shapes into an advanced rotational integration scheme. The integrator gives quantitative agreement with both analytic and approximate hydrodynamic theories for a number of model rigid bodies and works well at reproducing the solute dynamical properties (diffusion constants and orientational relaxation times) obtained from explicitly solvated simulations.

A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space

In this paper the contact problem between a rigid indenter of arbitrary shape and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenter and the distribution of contact pressure change. We wish to find the relations which link the pressure distribution, the resultant force on the indenter and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shape the problem may be solved numerically by using the Matrix Inversion Method (MIM), extended to viscoelastic case. In this method the boundary conditions are satisfied exactly at specified “matching points” (the mid-points of the boundary elements). It can be validated by comparing the numerical results to the analytic solutions in cases of a spherical asperity (loading and unloading) and a conical asperity (loading only). Finally, the method was implemented for a finite cylindrical shape with its curved face indenting the surface of the half-space. This last example shows the efficiency of the method in case of a prescribed penetration as well as a given normal load history.

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams–Bashforth scheme is employed for the advection terms and the Crank–Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.

The motion of solids in inviscid uniform vortical fields

We consider the general motion (translation and rotation) of a deformable or rigid body of arbitrary shape in a linear shear flow of an effectively inviscid and incompressible fluid possessing uniform vorticity. The ambient vorticity may be time-dependent. For two-dimensional configurations a solution with uniform vorticity is possible for all times and for three-dimensional, it is possible only initially or during a short time interval after the body is impulsively introduced into the fluid. General analytic expressions for the vortical force and moment exerted on an arbitrary moving body are presented. Bearing in mind applications for large non-spherical bubble dynamics, the general expressions for the hydrodynamic loads are further reduced for symmetric quadratic shapes such as two-dimensional ellipses or three-dimensional ellipsoids. The simplified expressions are given in terms of the body's added-mass tensor, its six velocities and the ambient vorticity. The few available degenerate solutions for cylinders and spheres are readily obtained as limiting cases.

On the Motion of Solids Through an Ideal Liquid: Approximated Equations for Many Body Systems

The problem of motion of many solids through an unbounded ideal liquid (inviscid and irrotational) is considered. A Lagrangian formulation of the equations of motion leads to a set of ordinary differential equations (ODEs) coupled to an elliptic partial differential equation (PDE) [H. Lamb, Hydrodynamics, 6th ed., Dover, New York, 1932]. Here, using a variational approach, an approximated solution for the PDE is presented, and the problem is reduced to the study of a system of ODEs. As a consequence one can get approximate forces and torques due to hydrodynamic interaction of rigid bodies of arbitrary shapes. Some examples are discussed at the end.

A rigid surface boundary element for soil-structure interaction analysis in the direct time domain

Many soil-structure interaction problems involve studies of single or multiple rigid bodies of arbitrary shape and soil media. The commonly used boundary element methods implement the equations of the rigid body in a form that depends on the particulars of the geometry and requires partitioning and condensation of the associated algebraic system of equations. The present work employs the direct time domain B-Spline BEM for 3D elastodynamic analysis and presents an efficient implementation of rigid bodies of arbitrary shape in contact with, or embedded in, elastic media. The formulation of a rigid surface boundary element introduced herein is suitable for direct superposition in the BEM system of algebraic equations. Consequently, solutions are computed in a single analysis step, eliminating, thus, the need for partitioning of the system of equations. Computational efficiency is also achieved due to the extremely sparse form of the associated coefficient matrices. The proposed element can be used for the modeling of single or multiple rigid bodies of arbitrary shape within the framework of the BEM method. The efficiency and general nature of the proposed element is demonstrated through applications related to the dynamic analysis of rigid surface and embedded foundations and their interaction with embedded rigid bodies of arbitrary shape.

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.