Balance Of Linear Momentum Research Articles

Implementation of standard penalty procedures for the solution of incompressible Navier–Stokes equations, employing the element-free Galerkin method

The present work introduces an alternative implementation of the element-free Galerkin method (EFGM) for the steady incompressible Navier–Stokes equations. The linear momentum balance and mass conservation equations have been developed on the basis of this global weak formulation. The weak forms of both equations have been coupled by means of two standard penalty procedures, which have been previously formulated and successfully employed in mesh-based numerical techniques such as the finite element method (FEM). These include the consistent penalty method (CPM) and reduced integration penalty method (RIPM). A detailed explanation concerning the characteristics inherent to the implementation of both penalty procedures in the EFGM based solutions for the incompressible Navier–Stokes equations, has also been provided. The resulting systems of equations have been adapted to the solution of different well-known incompressible flow benchmark problems. The feasibility and reliability of extending the implementation of these penalty procedures to the EFGM based solutions has been verified by comparison with the numerical techniques proposed and the results reported by other researchers. Results have revealed that this technique could be successfully used in the solution of Newtonian incompressible flow problems under the aforementioned penalty approaches.

Numerical simulations of nearly incompressible viscoelastic membranes

This work presents a novel numerical investigation of the dynamics of free-boundary flows of viscoelastic liquid membranes. The governing equation describes the balance of linear momentum, in which the stresses include the viscoelastic response to deformations of Maxwell type. A penalty method is utilized to enforce near incompressibility of the viscoelastic media, in which the penalty constant is proportional to the viscosity of the fluid. A finite element method is used, in which the slender geometry representing the liquid membrane, is discretized by linear three-node triangular elements under plane stress conditions. Two applications of interest are considered for the numerical framework provided: shear flow, and extensional flow in drawing processes.

An asymptotic homogenization approach to the microstructural evolution of heterogeneous media

In the present work, we apply the asymptotic homogenization technique to the equations describing the dynamics of a heterogeneous material with evolving micro-structure, thereby obtaining a set of upscaled, effective equations. We consider the case in which the heterogeneous body comprises two hyperelastic materials and we assume that the evolution of their micro-structure occurs through the development of plastic-like distortions, the latter ones being accounted for by means of the Bilby–Kröner–Lee (BKL) decomposition. The asymptotic homogenization approach is applied simultaneously to the linear momentum balance law of the body and to the evolution law for the plastic-like distortions. Such evolution law models a stress-driven production of inelastic distortions, and stems from phenomenological observations done on cellular aggregates. The whole study is also framed within the limit of small elastic distortions, and provides a robust framework that can be readily generalized to growth and remodeling of nonlinear composites. Finally, we complete our theoretical model by performing numerical simulations.

On the wave dispersion in microstructured solids

In this paper, elastic wave propagation in a one-dimensional micromorphic medium characterized by two internal variables is investigated. The evolution equations are deduced following two different approaches, namely using: (i) the balance of linear momentum and the Clausius–Duhem inequality, and (ii) an assumed Lagrangian functional (including a gyroscopic coupling) together with a variational principle. The dispersion relation is obtained and the possibility of the emerging band gaps is shown in such microstructured materials. Some numerical simulations are also performed in order to highlight the dispersive nature of the material under study.

Coupled effective stress analysis of insertion problems in geotechnics with the Particle Finite Element Method

This paper describes a computational framework for the numerical analysis of quasi-static soil-structure insertion problems in water saturated media. The Particle Finite Element Method is used to solve the linear momentum and mass balance equations at large strains. Solid-fluid interaction is described by a simplified Biot formulation using pore pressure and skeleton displacements as basic field variables. The robustness and accuracy of the proposal is numerically demonstrated presenting results from two benchmark examples. The first one addresses the consolidation of a circular footing on a poroelastic soil. The second one is a parametric analysis of the cone penetration test (CPTu) in a material described by a Cam-clay hyperelastic model, in which the influence of permeability and contact roughness on test results is assessed.

Modelling of environmentally assisted material degradation in the crack phase-field framework

The simulation of crack propagation was conducted with a diffusive crack model in a variational framework. More, the physically sound introduction of mass transport and coupling mechanisms due to environmentally assisted effects could be realised in this framework. The objective consists of the application of the phase-field approach towards the simulation of environmentally assisted material degradation with the advantage of a non-required predefined crack path and a mesh-independent non-local formulation that facilitates the damage evolution with respect to material softening. The sharp crack is regularised by the introduction of a phase-field order parameter leading to a diffusive crack formulation. Besides the equations originating from the linear momentum balance an additional evolution equation for the crack phase-field is introduced. Furthermore, mass transport is simulated by a diffusion equation. The description delivered by the variational phase-field framework is able to simulate crack propagation according to published numerical test cases. Additionally, the calculation of stress intensity factors is possible as well as crack resistance curves that describe stable crack propagation.

Efficient implementation of energy conservation for higher order finite elements with variational integrators

In this paper, we introduce a variational integrator for higher order finite element models. The solution quality of finite element simulation strongly depends on the approximation in time and space. A variational integrator is robust due to conservation of both the balance of total linear momentum and the balance of total angular momentum. It also binds the error in the balance of total energy. We extend the variational integrator to an integrator that conserves the balance of total energy as well. However, the approximation in time cannot mend the approximation error in space. Therefore, we use a higher order approximation in space to obtain a better solution compared to the real live model that should be simulated. We show different numerical examples with Dirichlet and Neumann boundaries. For the Dirichlet boundaries we use the Lagrange multiplier method and for the Neumann boundaries we introduce the forces on the nodes by constant pressure.

Modelling of strain softening materials based on equivalent damage force

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor.The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete.The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density.

Simulating the hydrodynamic response of a floater–net system in current and waves

We present a novel numerical model for simulating current and wave interaction with a floater–net system. The main contribution of the paper is the integration of the floater motion and the fluid–structure interaction analysis of the net structure in the same modelling framework via the computational fluid dynamic approach. The sinker and the mooring lines were not directly resolved, but their effects were partially modelled. The model couples a hydrodynamic solver, a rigid body motion solver, a mesh motion solver and a structural solver in a segregated manner. In the numerical model, the net structure was modelled as a set of dynamic porous zones. A lumped mass model was coupled with it to realize fluid–structure interaction analysis for the net structure. The floater was treated as a rigid body, which was resolved by the body-fitted computational mesh in the fluid domain. The motion equation for the floater was set up based on the principle of linear and angular momentum balance. Different motion integration schemes were implemented and tested in the numerical model. The numerical model was validated against three sets of available experimental data in the open literature. The first set of validation cases treated the floater motion in regular waves. The second set of validation cases focused on the fluid–structure interaction analysis of the net structure. The final one was related to the whole floater–net system in regular waves, and combined current and wave condition.

Finite Element Framework for Computational Fluid Dynamics in FEBio.

The mechanics of biological fluids is an important topic in biomechanics, often requiring the use of computational tools to analyze problems with realistic geometries and material properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities. This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal-order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBio could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid-structure interactions (FSIs).

Phase‐Field Models for the Failure of Anisotropic Continua

AbstractThis study presents a phase‐field approach for an anisotropic continuum to model fracture of biological tissues and fiber‐reinforced composites. We start with the continuous formulation of the variational principle for the multi‐field problem manifested through the deformation map and the crack phase‐field at finite strains which leads to the Euler–Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the anisotropic crack phase‐field and the balance of linear momentum. In addition, we propose a novel energy‐based anisotropic failure criterion which regulates the evolution of the crack phase‐field. The coupled problem is solved using a one‐pass operator‐splitting algorithm composed of a mechanical predictor step and a crack evolution step. Representative numerical examples are devised for crack initiation and propagation in carbon‐fiber‐reinforced polymerg composites. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results qualitatively capture the effect of anisotropy in stiffness and strength. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlocal balance of angular momentum from stress field analysis

AbstractNonlocal theories drop the principle of local action such that the stress field in the contiguity of a continuum point is represented by a nonlinear tensor‐function. Therefore, its power series expansion has higher order terms and the balance of linear momentum becomes nonlocal, see [1, 2]. Here, we focus on the balance of angular momentum by considering up to quadratic terms in the approximation of the stress field. This yields nonlinear tractions on the surface of spatial unit cells, where an algebraic formula for the couple stress tensor can be found. However, couple stress is an auxiliary quantity, which is usually motivated as work conjugated to curvature. With help of some algebra in ℝ3 × ℝ3 × ℝ3 it is found, that the curl of the stress field rules the nonlocal balance of angular momentum. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computational inelasticity for loading conditions on multiple time scales by adaptive step size control

AbstractThis contribution proposes an algorithm based on adaptive step size control for the simulation of inelastic solids and structures undergoing loading conditions at multiple time scales. Adaptivity in time integration of viscoelastic constitutive laws is directed by an refinement indicator which is constructed from integrators of different order, here a fourth‐order Runge‐Kutta (RK) method and linear Backward‐Euler. The key novel aspect is that by virtue of an recently established consistency condition the higher order methods, p ≥ 2, can achieve their full nominal order without fulfilling the weak form of balance of linear momentum in the RK stages, but only at the end of the time interval. A representative numerical example illustrates the performance of the present adaptive method and underpins the computational savings compared with uniform time step sizes. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Statistical mechanics of transport processes in active fluids: Equations of hydrodynamics.

The equations of hydrodynamics including mass, linear momentum, angular momentum, and energy are derived by coarse-graining the microscopic equations of motion for systems consisting of rotary dumbbells driven by internal torques. In deriving the balance of linear momentum, we find that the symmetry of the stress tensor is broken due to the presence of non-zero torques on individual particles. The broken symmetry of the stress tensor induces internal spin in the fluid and leads us to consider the balance of internal angular momentum in addition to the usual moment of momentum. In the absence of spin, the moment of momentum is the same as the total angular momentum. In deriving the form of the balance of total angular momentum, we find the microscopic expressions for the couple stress tensor that drives the spin field. We show that the couple stress contains contributions from both intermolecular interactions and the active forces. The presence of spin leads to the idea of balance of moment of inertia due to the constant exchange of particles in a small neighborhood around a macroscopic point. We derive the associated balance of moment of inertia at the macroscale and identify the moment of inertia flux that induces its transport. Finally, we obtain the balances of total and internal energy of the active fluid and identify the sources of heat and heat fluxes in the system.

Numerical study of temperature effects on the poro-viscoelastic behavior of articular cartilage.

This paper presents a new approach to study the effects of temperature on the poro- elastic and viscoelastic behavior of articular cartilage. Biphasic solid-fluid mixture theory is applied to study the poro-mechanical behavior of articular cartilage in a fully saturated state. The balance of linear momentum, mass, and energy are considered to describe deformation of the solid skeleton, pore fluid pressure, and temperature distribution in the mixture. The mechanical model assumes both linear elastic and viscoelastic isotropic materials, infinitesimal strain theory, and a time-dependent response. The influence of temperature on the mixture behavior is modeled through temperature dependent mass density and volumetric thermal strain. The fluid flow through the porous medium is described by the Darcy's law. The stress-strain relation for time-dependent viscoelastic deformation in the solid skeleton is described using the generalized Maxwell model. A verification example is presented to illustrate accuracy and efficiency of the developed finite element model. The influence of temperature is studied through examining the behavior of articular cartilage for confined and unconfined boundary conditions. Furthermore, articular cartilage under partial loading condition is modeled to investigate the deformation, pore fluid pressure, and temperature dissipation processes. The results suggest significant impacts of temperature on both poro- elastic and viscoelastic behavior of articular cartilage.

Peristaltic axisymmetric flow of a Bingham fluid

We model the peristaltic flow of a Bingham fluid in a tube in lubrication approximation. Following the procedure developed in Fusi et al. (2015a) we derive the rigid plug equation using an integral formulation for the balance of linear momentum, modelling the unyielded domain as an evolving non-material volume. The mathematical problem is formulated for the yielded and unyielded part and appropriate boundary conditions are established at the pipe walls and at the yield surface. The zero order approximation leads to a system formed by an integral equation and an algebraic equation for the yield surface and for the plug velocity (which is uniform in space), respectively. Because of the integral approach adopted in the unyielded part of the flow, the leading order approximation does not give rise to the lubrication paradox. The problem is solved numerically and an analytical solution is found when the oscillating wall is given as a small perturbation of the uniform wall.

Simple iterative procedure for the thermal–mechanical analysis of continuous casting processes, using the element-free Galerkin method

ABSTRACTThe present work has been conducted to develop a straightforward procedure for solving the thermal–mechanical problem in a conventional continuous casting (CC) process using the element free Galerkin method. The linear momentum balance equation has been developed on the basis of this global weak formulation. The model has been solved under a nodes-fixed mixed Eulerian–Lagrangian description to estimate the mechanical behavior of a round billet during the CC process in the mold region. The proposed approach is developed in a predictor–corrector scheme, which provides a solution involving simultaneously all the materials configuration downwards the mold without the requirement of a viscoplastic tangent operator. The results have revealed that this technique could be used successfully in the modeling of continuous casting thermal–mechanical problems.

Structure-preserving time integration of non-isothermal finite viscoelastic continua related to variational formulations of continuum dynamics

A structure-preserving time integration scheme is a modified standard time stepping scheme, which fulfills the balance laws of a dynamical problem also in a discrete setting. Hence, such a scheme represents a space-time discretization of a continuum with all of its central properties. This paper deals with a special second-order accurate structure-preserving time integration of thermo-mechanical couplings in a moving continuum, satisfying all discrete thermodynamical balance laws in contrast to the underlying second-order accurate midpoint rule. More accurately, we consider isotropic heat conduction and the Gough–Joule effect in moving continua. The considered time evolution is described by five differential equations with respect to five independent variables, including the entropy field of the continuum. The bodies in the numerical examples are subjected to Dirichlet- or Neumann boundary conditions in the displacement as well as the temperature field. Although this structure-preserving time integrator is based on a time discrete spatially weak finite element formulation, it fulfills the same balance laws as the underlying five differential equations. Namely, in addition to the balances of entropy, linear momentum and angular momentum, also the balances of total energy and Lyapunov’s function are fulfilled in the discrete case.

A Thermo-Poro-Viscoelastic Model for the Behavior of Polymeric Porous Separators

This paper presents a continuum formulation for thermo-viscoelastic behavior of fully saturated porous media. Two-phase mixture with solid and fluid phases is considered. Governing and balance equations of the mixture, including the balance of mass, balance of linear momentum and balance of energy are presented. The displacement of the solid skeleton, pore fluid pressure, and temperature of the mixture are considered as independent state variables. The stress-strain relationship of the solid skeleton is described by a linear viscoelastic model. The fluid flow is described by the Darcy's law. Numerical examples are presented to show the performance of the model.

Submesoscale surface fronts and filaments: secondary circulation, buoyancy flux, and frontogenesis

Problems are posed and solved for upper-ocean submesoscale density fronts and filaments in the presence of surface wind stress and the associated boundary-layer turbulent mixing, their associated geostrophic and secondary circulations and their instantaneous buoyancy fluxes and frontogenetic evolutionary tendencies in both velocity and buoyancy gradients. The analysis is diagnostic rather than prognostic, and it is based on a momentum-balanced approximation that assumes the ageostrophic acceleration is negligible, although the Rossby number is finite and ageostrophic advection is included, justified by the quasi-steady, coherent-structure flow configurations of fronts and filaments. Across a wide range of wind and buoyancy-gradient parameters, the ageostrophic secondary circulation for a front is a single overturning cell with downwelling on the dense side, hence with a positive (restratifying) vertical buoyancy flux. For a dense filament the circulation is a double cell with central downwelling and again positive vertical buoyancy flux. The primary explanation for these secondary-circulation cells is a ‘turbulent thermal wind’ linear momentum balance. These circulation patterns, and their associated frontogenetic tendencies in both the velocity and buoyancy gradients, are qualitatively similar to those due to the ‘classical’ mechanism of strain-induced frontogenesis. For linear solutions, the secondary circulation and frontogenesis are essentially independent of wind direction, but in nonlinear solutions ageostrophic advection provides a strong intensification of the peak vertical velocity, while generally preserving the ageostrophic circulation pattern, when the Rossby number is order one and the wind orientation relative to the frontal axis is favourable. At large Rossby number the solution procedure fails to converge, with an implication of a failure of existence of wholly balanced circulations.