Nodal Contact Forces Research Articles

Multibody dynamic modeling of the behavior of flexible instruments used in cervical cancer brachytherapy.

The steep radiation dose gradients in cervical cancer brachytherapy (BT) necessitate a thorough understanding of the behavior of afterloader source cables or needles in the curved channels of (patient-tailored) applicators. The purpose of this study is to develop and validate computer models to simulate: (1) BT source positions, and (2) insertion forces of needles in curved applicator channels. The methodology presented can be used to improve the knowledge of instrument behavior in current applicators and aid the development of novel (3D-printed) BT applicators. For the computer models, BT instruments were discretized in finite elements. Simulations were performed in SPACAR by formulating nodal contact force and motion input models and specifying the instruments' kinematic and dynamic properties. To evaluate the source cable model, simulated source paths in ring applicators were compared with manufacturer-measured source paths. The impact of discrepancies on the dosimetry was estimated for standard plans. To validate needle models, simulated needle insertion forces in curved channels with varying curvature, torsion, and clearance, were compared with force measurements in dedicated 3D-printed templates. Comparison of simulated with manufacturer-measured source positions showed 0.5-1.2mm median and<2.0mm maximum differences, in all but one applicator geometry. The resulting maximum relative dose differences at the lateral surface and at 5mm depth were 5.5% and 4.7%, respectively. Simulated insertion forces for BT needles in curved channels accurately resembled the forces experimentally obtained by including experimental uncertainties in the simulation. The models developed can accurately predict source positions and insertion forces in BT applicators. Insights from these models can aid novel applicator design with improved motion and force transmission of BT instruments, and contribute to the estimation of overall treatment precision. The methodology presented can be extended to study other applicator geometries, flexible instruments, and afterloading systems.

Solving contact forces with the competition between potential energy and entropy in elastic mechanics

Energy and entropy are competitors in thermodynamics, as an increase in one usually leads to a decrease in the other. In this article, we show that competition between potential energy and entropy also exists in contact problems, which is revealed in the process of solving contact forces in the finite element model of elastic contact problems based on the principle of minimum potential energy and the principle of maximum entropy. First, we discretize the contact problem with finite elements and model it as a quadratic programming problem with the potential energy as the objective and non-penetration conditions as the constraints. Second, by normalizing the Lagrange multipliers—the potential nodal contact forces—in the Lagrangian duality, we obtain an explicit formulation of the displacements as expressed by the normalized potential nodal contact forces. Third, analogous to the concepts in statistical physics, we consider each normalized potential nodal contact force as the probability of the node occupying a microstate, and with the maximum entropy principle, we obtain an explicit formulation of the normalized potential nodal contact forces as expressed by the displacements. Fourth, we construct an iterative procedure for solving the contact forces based on the previous two formulations. The initial value of each normalized nodal contact force, which is one divided by the number of potential contact nodes, is used to calculate the displacements in the first iteration, the iterations continue until a termination condition that is based on the monotonicity of the potential energy is met. Finally, we provide some examples to verify the iterative algorithm and the competition between the potential energy and entropy in the iterative procedure, which takes the form of a decrease in entropy and an increase in potential energy.

Computing an upper bound on contact stress with surrogate duality

We present a method for computing an upper bound on the contact stress of elastic bodies. The continuum model of elastic bodies with contact is first modeled as a constrained optimization problem by using finite elements. An explicit formulation of the total contact force, a fraction function with the numerator as a linear function and the denominator as a quadratic convex function, is derived with only the normalized nodal contact forces as the constrained variables in a standard simplex. Then two bounds are obtained for the sum of the nodal contact forces. The first is an explicit formulation of matrices of the finite element model, derived by maximizing the fraction function under the constraint that the sum of the normalized nodal contact forces is one. The second bound is solved by first maximizing the fraction function subject to the standard simplex and then using Dinkelbach's algorithm for fractional programming to find the maximum--since the fraction function is pseudo concave in a neighborhood of the solution. These two bounds are solved with the problem dimensions being only the number of contact nodes or node pairs, which are much smaller than the dimension for the original problem, namely, the number of degrees of freedom. Next, a scheme for constructing an upper bound on the contact stress is proposed that uses the bounds on the sum of the nodal contact forces obtained on a fine finite element mesh and the nodal contact forces obtained on a coarse finite element mesh, which are problems that can be solved at a lower computational cost. Finally, the proposed method is verified through some examples concerning both frictionless and frictional contact to demonstrate the method's feasibility, efficiency, and robustness.

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.

Modeling and simulation of a frictional translational joint with a flexible slider and clearance

A finite element method for the dynamic description of multibody systems with a frictional translational joint is presented and discussed in this work. By considering the slider deformation and the clearance size between the slider and the guide, Coulomb and Stribeck models are used to describe friction forces. The geometric constraints of the translational joints are treated as multi-unilateral constraints, and the impacts between the slider and guide are neglected when the clearance sizes are very small. A model of contact forces is established based on the penalty method and a criterion is proposed to evaluate the influence of clearance size on nodal contact forces. Time-varying contact states of nodes are determined by applying two contact detection algorithms for stick and slip, respectively. Based on kineto-elasto-dynamic method and varying penalty factors, the method avoids convergence difficulties and enforces the accuracy. Numerical results reveal the capability of the new method and the influences of small deformations of the slider, clearance size and friction forces on the dynamic response of the multibody systems with a translational joint.

Mixed Finite Element Method for Static and Dynamic Contact Problems with Friction and Initial Gaps

A novel mixed finite element method is proposed for static and dynamic contact problems with friction and initial gaps. Based on the characteristic of local nonlinearity for the problem, the system of forces acting on the contactor is divided into two parts: external forces and contact forces. The displacement of structure is chosen as the basic variable and the nodal contact force in contact region under local coordinate system is selected as the iteration variable to confine the nonlinear iteration process in the potential contact surface which is more numerically efficient. In this way, the sophisticated contact nonlinearity is revealed by the variety of the contact forces which are determined by the external load and the contact state stick, slip, or separation. Moreover, in the case of multibody contact problem, the flexibility matrix is symmetric and sparse; thus, the iterative procedure becomes easily carried out and much more economical. In the paper, both the finite element formulations and the iteration process are given in detail for static and dynamic contact problems. Four examples are included to demonstrate the accuracy and applicability of the presented method.

Numerical study of contact forces for crack closure analysis

Plasticity induced crack closure (PICC) has been widely studied using numerical models. Different numerical parameters can be considered to quantify the opening level, namely one based on the analysis of contact stresses at minimum load. A modified version of this parameter is proposed here, based on nodal contact forces instead of contact stresses. The predictions were found to be similar to those obtained from the contact status of 2nd node behind crack tip. The PICCcontact parameter was also found to be very consistent and adequate for parametric studies of the influence of different physical parameters. The contributions to the opening stress intensity factor of different points along crack flank were found to strongly decrease with distance to crack tip. The cumulative Kopen between the crack tip and a distance of 0.1mm was found to vary from 30% to 100%, increasing with stress ratio, R. Finally, a K solution was developed for punctual forces applied on crack flank and compared with a literature solution for infinite plates. A good agreement was found for plane strain state but significant differences of about 10% were found for plane stress state.

Implementation of Fictitious Crack Model Using Contact Finite Element Method for the Crack Propagation in Concrete under Cyclic Load

The mixed freedom finite element method proposed for contact problems was extended to simulate the fracture mechanics of concrete using the fictitious crack model. Pairs of contact points were set along the potential developing path of the crack. The displacement of structure was chosen as the basic variable, and the nodal contact force in contact region under local coordinate system was selected as the iteration variable to confine the nonlinear iteration process in the potential contact surface which is more numerically efficient. The contact forces and the opening of the crack were obtained explicitly enabling the softening constitutive relation for the concrete to be introduced conveniently by the fictitious crack model. According to the states of the load and the crack, the constitutive relation of concrete under cyclic load is characterized by six contact states with each contact state denoting its own displacement-stress relation. In this paper, the basic idea of the mixed freedom finite element method as well as the constitutive relation of concrete under cyclic load is presented. A numerical method was proposed to simulate crack propagation process in concrete. The accuracy and capability of the proposed method were verified by a numerical example against experiment data.

Non-linear strain–displacement equations exactly representing large rigid-body motions. Part III. Analysis of TM shells with constraints

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko–Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress–strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples.

Development of a Finite Element Contact Analysis Algorithm to Pass the Patch Test

A finite element contact analysis algorithm has to pass the so-called patch test. In the conventional one-pass approach, the virtual work due to contact force is evaluated by concentrated contact force and corresponding virtual displacement. This means that the contact force at the contact surface between deformable bodies is not transmitted appropriately in the virtual work sense. This paper proposes a new algorithm based on the one-pass approach, which evaluates the virtual work due to contact force by integrating it on the contact surfaces. Furthermore, the nodal contact pressure in the master contact surface is transmitted from that in the slave contact surface by projecting the master node onto the slave element. In this way, the proposed algorithm is capable of correctly evaluating the equivalent nodal contact force and passing the contact patch test. Two numerical examples verify the effectiveness of the new algorithm.

A sophisticated node-pair model for interface problems

Based on the virtual work principle of interface problems, this paper presents an innovative finite element solution for interface problems by mixing nodal contact forces with nodal displacements. The proposed mixed finite element solution can overcome some numerical difficulties encountered in the analysis of contact problems in geomechanics. These difficulties include contact of large area, non-smooth contact, alleviation of ill-condition and treatment of rigid displacements. The well-posed issue of the formulation and a criterion for judging collapse of the system are also discussed in detail. The paper then presents two simple examples and a very challenging example from the shiplock of the Three Gorges Project in construction. These examples illustrate that the proposed mixed finite element solution is robust and suitable for solving interface problems encountered in geomechanics.

FINITE ELEMENT ANALYSIS OF FRICTIONALLY EXCITED THERMOELASTIC INSTABILITY

The frictional heat generated during braking causes thermoelastic distortion that modifies the contact pressure distribution. If the sliding speed is sufficiently high, this can lead to frictionalfy excited thermoelastic instability, characterized by major nonuniformi-ties in pressure and temperature. In automotive applications, a particular area of concern is the relation between thermoelasticalfy induced hot spots in the brake disks and noise and vibration in the brake system. The critical sliding speed can be found by examining the conditions under which a perturbation in the temperature and stress fields can grow in time. The growth has exponential character, and subject to certain restrictions, the growth rate b is found to be real. The critical speed then corresponds to a condition at which b = 0 and hence at which there is a steady-state solution involuing nonuniform contact pressure. We first treat the heat sources Q at the contact nodes as given and use standard finite element analysis (FEA) to determine the corresponding nodal contact forces P. The heat balance equation Q =fVP, where f is the coefficient of friction, then defines a linear eigenvalue problem for the critical speed V. The method is found to give good estimates for the critical speed in test cases with a relatively coarse mesh. It is generally better conditioned and more computationally efficient than a direct finite element simulation of the system in time. Results are presented for several examples related to automotive practice and show that theflexural rigidity of the friction pad assembly has a major effect on the critical speed.

Application of contact-friction interface element to tunnel excavation in faulted rock

A simple contact-friction interface element for two-dimensional models is developed here. This model takes nodal contact stresses instead of nodal contact forces as unknowns. The element can simulate frictional slippage, decoupling and re-bonding of two bodies initially mating at a common interface. The method is based on the Finite Element Method and load incremental theory. The geometric and static constraint conditions on contact surfaces are treated as additional conditions and are included in stiffness equations. Since the stiffness matrix is unsymmetric, symmetrizing techniques are discussed in this paper. Consequently, a symmetrical stiffness matrix of the constraint element is established. The new element has the advantages of easy implementation into standard finite element programs and high speed for convergence. As an example of its application, the excavation of a tunnel with soft media and joint interfaces is simulated. The results obtained demonstrate the feasibility of using this model to perform analyses of discontinuties in rock mass, such as faults, shear zones and joints.

On geometrically nonlinear contact problems with friction

The analysis of quasi-static contact problems applying an ‘updated’ respectively a ‘total Lagrangian’ formulation is extended in a way that frictional effects according to Coulomb's law are being incorporated into the incremental procedure which is closely related to the unit load method. Further novel aspects are to be found in the iterative computation of the tangential components of the stress distribution in the contact area due to various boundary conditions. The second part of the paper deals with the application of the nonlinear programming technique to problems with unilateral constraints. Using standard finite element techniques an ‘augmented Lagrangian’ is defined for a predeformed compressed rod. Neglecting frictional effects, the contact zones and the nodal contact forces between the deformed rod and lateral rigid objects are described by the Kuhn-Tucker conditions.