Tod A. Laursen

An augmented lagrangian treatment of contact problems involving friction

A framework is presented within which the method of augmented Lagrangians is readily applied to problems involving contact with friction. This method, which has enjoyed considerable success in the treatment of constrained minimization problems, has been previously applied to problems involving incompressible flow, incompressible elasticity of solids and even frictionless contact. An additional challenge to the method is provided by frictional contact problems governed by a Coulomb law, due to the special form taken by the frictional constraint. This paper describes a new extension of the augmented Lagrangian technique to frictional problems which is well-suited to finite element implementation. The proposed treatment inherits the traditional advantages of augmented Lagrangian techniques over penalty methods; namely, decreased ill-conditioning of governing equations, and essentially exact satisfaction of constraints with finite penalties. A set of numerical examples is presented in which the utility of the method is demonstrated even in the presence of finite deformations and inelasticity.

Computational Contact and Impact Mechanics

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A study of the mechanics of microindentation using finite elements

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DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS

In this paper the finite element method is used to explore the mechanics of the microindentation process. In the simulations discussed, aluminum and silicon are investigated both in their bulk forms and in thin film-substrate combinations. Among the quantities readily computed using this approach and given in this paper are hardness (computed using actual contact area), contact stiffness, effective composite modulus, and surface profile under load. Importantly, this investigation builds on previous work by providing a more critical examination of the amount of pileup (or sink-in) around the indenter in the fully loaded configuration, as well as the variation of the actual contact area during indenter withdrawal. A key conclusion of this study is that finite element simulations do not support the widely used assumption of constancy of area during unloading (for either bulk materials or thin film systems). Furthermore, the amount of pileup or sink-in can be appreciable. The implication of these findings is that for many situations the commonly used straight-line extrapolation of a plastic depth may render an estimate for the contact area that is quite distinct from the actual area. This assertion is demonstrated herein through comparison of hardnesses calculated using actual contact area with values calculated using the straight-line extrapolation of plastic depth.

The Micromechanical Environment of Intervertebral Disc Cells Determined by a Finite Deformation, Anisotropic, and Biphasic Finite Element Model

This paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in finite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacrifices the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations.

Algorithmic symmetrization of coulomb frictional problems using augmented lagrangians

Cellular response to mechanical loading varies between the anatomic zones of the intervertebral disc. This difference may be related to differences in the structure and mechanics of both cells and extracellular matrix, which are expected to cause differences in the physical stimuli (such as pressure, stress, and strain) in the cellular micromechanical environment. In this study, a finite element model was developed that was capable of describing the cell micromechanical environment in the intervertebral disc. The model was capable of describing a number of important mechanical phenomena: flow-dependent viscoelasticity using the biphasic theory for soft tissues; finite deformation effects using a hyperelastic constitutive law for the solid phase; and material anisotropy by including a fiber-reinforced continuum law in the hyperelastic strain energy function. To construct accurate finite element meshes, the in situ geometry of IVD cells were measured experimentally using laser scanning confocal microscopy and three-dimensional reconstruction techniques. The model predicted that the cellular micromechanical environment varies dramatically between the anatomic zones, with larger cellular strains predicted in the anisotropic anulus fibrosus and transition zone compared to the isotropic nucleus pulposus. These results suggest that deformation related stimuli may dominate for anulus fibrosus and transition zone cells, while hydrostatic pressurization may dominate in the nucleus pulposus. Furthermore, the model predicted that micromechanical environment is strongly influenced by cell geometry, suggesting that the geometry of IVD cells in situ may be an adaptation to reduce cellular strains during tissue loading.

Optimization study and heat transfer comparison of staggered circular and elliptic tubes in forced convection

Abstract A major drawback of most implicit algorithmic treatments of Coulomb frictional contact problems is the nonsymmetry of the algorithmic tangent operator, which emanates from the nonassociativity of the slip (flow) rule. Reliable algorithms leading to symmetric equation systems are highly desirable, especially in three-dimensional problems where the computational cost is dominated by equation solving. In this paper such an algorithm symmetrization is proposed. The method is derived from the augmented Lagrangian treatment of friction given by Simo and Laursen (Comput. & Structures 42 (1992) 97–116), and exploits the iterative nature of the classical method of multipliers. More specifically, symmetrization is achieved by a modified multiplier update scheme in which all nonassociativity is removed from the solution phase and placed in the multiplier update phase. The result is an algorithm involving only the solution of symmetric equations, which maintains the attributes of an augmented Lagrangian method.

A framework for development of surface smoothing procedures in large deformation frictional contact analysis

Abstract In this study, a two-dimensional (2-D) heat transfer analysis was performed in circular and elliptic tube heat exchangers. The finite element method was used to discretize the fluid flow and heat transfer governing equations and a 2-D isoparametric, four-noded, linear element was implemented for the finite element analysis program, FEAP (O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1, McGraw-Hill, London, 1989, Chapter 15). The numerical results for the equilateral triangle staggering configuration, obtained with the new element were then validated qualitatively by means of direct comparison to previously published experimental results for circular tubes heat exchangers (G. Stanescu, A.J. Fowler, A. Bejan, Int. J. Heat Mass Transfer 39 (2) (1996) 311–317). Next, a numerical geometric optimization was conducted to maximize the total heat transfer rate between the given volume and the given external flow both for circular and elliptic arrangements, for general staggering configurations. The results are reported for air in the range 300⩽ReL⩽800, where L is the swept length of the fixed volume. Circular and elliptical arrangements with the same flow obstruction cross-sectional area were compared on the basis of maximum total heat transfer. The effect of ellipses eccentricity was also investigated. A relative heat transfer gain of up to 13% is observed in the optimal elliptical arrangement, as compared to the optimal circular one. The heat transfer gain, combined with the relative pressure drop reduction of up to 25% observed in previous studies (H. Brauer, Chem. Process Eng., August (1964) 451–460; S.N. Bordalo, F.E.M. Saboya, Determinacao experimental dos coeficientes de perda de carga em trocadores de calor de tubos circulares e elipticos aletados, in: Proceedings of the 13th COBEM, Congresso Brasileiro de Engenharia Mec a nica (in Portuguese), Belo Horizonte, Brasil, 1995) show that the elliptical arrangement has the potential for a considerably better overall performance than the traditional circular one.

Large Deformation Finite Element Analysis of Micropipette Aspiration to Determine the Mechanical Properties of the Chondrocyte

This paper presents a variational framework and numerical implementation strategy for contact smoothing algorithms in large deformation finite element analysis. Since traditional approaches in contact mechanics enforce kinematic constraints defined in terms of faceted surfaces introduced by finite element discretizations, large sliding problems in particular tend to induce nonphysical oscillations in contact forces and nonlinear equation solving difficulties associated with the discontinuities in the contact surface geometry. The formulation presented herein is shown to circumvent these issues, improving the robustness of the numerical procedure and eliminating many of the contact force oscillations given by more traditional procedures.

Energy consistent algorithms for dynamic finite deformation plasticity

Chondrocytes, the cells in articular cartilage, exhibit solid-like viscoelastic behavior in response to mechanical stress. In modeling the creep response of these cells during micropipette aspiration, previous studies have attributed the viscoelastic behavior of chondrocytes to either intrinsic viscoelasticity of the cytoplasm or to biphasic effects arising from fluid–solid interactions within the cell. However, the mechanisms responsible for the viscoelastic behavior of chondrocytes are not fully understood and may involve one or both of these phenomena. In this study, the micropipette aspiration experiment was modeled using a large strain finite element simulation that incorporated contact boundary conditions. The cell was modeled using finite strain incompressible and compressible elastic models, a two-mode compressible viscoelastic model, or a biphasic elastic or viscoelastic model. Comparison of the model to the experimentally measured response of chondrocytes to a step increase in aspiration pressure showed that a two-mode compressible viscoelastic formulation accurately captured the creep response of chondrocytes during micropipette aspiration. Similarly, a biphasic two-mode viscoelastic analysis could predict all aspects of the cell’s creep response to a step aspiration. In contrast, a biphasic elastic formulation was not capable of predicting the complete creep response, suggesting that the creep response of the chondrocytes under micropipette aspiration is predominantly due to intrinsic viscoelastic phenomena and is not due to the biphasic behavior.