A time integration scheme with energy–momentum conservation for a shell formulation with arbitrary geometric and material non-linearities

The paper is concerned with a time integration scheme which conserves energy, momentum and angular momentum for shells exhibiting arbitrary non-linearities in the strain–displacement relations together with a possible non-linear constitutive behaviour and displacement-dependent loading. The formulation is general and can apply to any shell formulation. However, we derive the equations for the specific case of the so-called seven-parametric shell theory which is characterized by a quadratic displacement field over the shell thickness. Numerical examples of large overall motion of shells are provided showing the main features of the algorithm.     

Transportation of neutral solute in deformable,anisotropic, soft tissue

We present a three-dimensional framework for the transport of neutral solute through soft biological tissue, such as the articular cartilage, under different mechanical loading conditions. The tissue is modelled as a mixture of three phases, a hyperelastic, anisotropic solid matrix, an incompressible inviscid fluid and a solute. The formulation accounts for the anisotropic interaction between the solid and the interstitial water and incorporates the interaction between the solute and the solid matrix. Under the proper circumstances, our model simplifies to biphasic theory and classical convection diffusion.We find that the pressure gradient variation plays an important role in the neutral solute transport when the soft tissue is dynamically loaded, because of the low concentration of solute present in cartilage. A linear function is suggested for the ratio of friction coefficient between the solute and solid to the friction coefficient between the interstitial water and solid matrix. After validating our model with the experiments of Quinn et al. [T.M. Quinn, C. Studer, A.J. Grodzinsky, J.J. Meister, Preservation and analysis of nonequilibrium solute concentration distributions within mechanically compressed cartilage explants, J. Biochem. Biophys. Methods 52 (2) (2002) 83–95], a numerical study of different dynamic loading conditions on solute transport is carried out.     

Numerical simulation of two-phase flow in deformable porous media: Application to carbon dioxide storage in the subsurface

In this paper, conceptual modeling as well as numerical simulation of two-phase flow in deep, deformable geological formations induced by CO₂ injection are presented. The conceptual approach is based on balance equations for mass, momentum and energy completed by appropriate constitutive relations for the fluid phases as well as the solid matrix. Within the context of the primary effects under consideration, the fluid motion will be expressed by the extended Darcy's law for two phase flow. Additionally, constraint conditions for the partial saturations and the pressure fractions of carbon dioxide and brine are defined. To characterize the stress state in the solid matrix, the effective stress principle is applied. Furthermore, the interaction of fluid and solid phases is illustrated by constitutive models for capillary pressure, porosity and permeability as functions of saturation. Based on this conceptual model, a coupled system of nonlinear differential equations for two-phase flow in a deformable porous matrix (H²M model) is formulated. As the displacement vector acts as primary variable for the solid matrix, multiphase flow is simulated using both pressure/pressure or pressure/saturation formulations. An object-oriented finite element method is used to solve the multi-field problem numerically. The capabilities of the model and the numerical tools to treat complex processes during CO₂ sequestration are demonstrated on three benchmark examples: (1) a 1-D case to investigate the influence of variable fluid properties, (2) 2-D vertical axi-symmetric cross-section to study the interaction between hydraulic and deformation processes, and (3) 3-D to test the stability and computational costs of the H²M model for real applications.     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

Numerical modeling of the mixing flow of second-order fluids with helical ribbon impellers

Rheology is known to have a strong impact on the flow behavior and the power consumption of mechanically agitated tanks. In the case of viscoelastic fluids, the role of elasticity on power draw has yet to be elucidated, though recent studies with helical ribbon impellers indicate that elasticity increases torque. The objective of this paper is to show that, in the case of second-order fluids, the use of a simple constitutive equation derived from a second-order retarded-motion expansion succeeds in predicting a rise in power draw owing to elasticity. The equations of change governing fluid flow are solved using a finite element method combined with an augmented Lagrangian method for the treatment of the non-linear constitutive equation. We show, in particular, how the underlying non-linear tensor equations can be solved directly using a spectral decomposition of the related matrix operator. First, the numerical methodology is explained. Next, simulation results are presented in the case of a tank provided with a helical ribbon. These results are then compared with experimental data. A rise of power draw, similar in shape but smaller than that measured experimentally, may be noted when elasticity is increased.     

弹塑性梁系统的动力学特性研究

研究了弹塑性梁系统的动力学特性.从弹塑性梁的非线性本构关系出发,同时考虑几何非线性,用虚功原理建立单个梁的动力学变分方程,利用假设模态法离散.在此基础上引入运动学约束关系,建立了弹塑性梁系统的刚-柔耦合动力学方程.对重力作用下的柔性单摆和双摆数值仿真结果表明,塑性应变引起横向变形绝对值增大和横向振动振幅衰减,在角加速度突变时塑性效应最为显著.     

针穿刺软组织变形有限元仿真

为了提高机器人辅助针穿刺精度,提出一种对软组织变形分析的准静态有限元法.该方法基于针受力下软组织变形机理,将软组织变形的连续动态过程分解成离散的准静态过程;采用重叠单元法在ANSYS中建立了软组织变形的二维和三维准静态有限元模型,并进行了仿真实验.最后通过实验结果验证了该方法的有效性和可行性.     

Approximation of time-dependent,multi-component,viscoelastic fluid flow

In this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow. The Oldroyd B constitutive equation is used to model the viscoelastic stress. For the discretization, time derivatives are replaced by backward difference quotients, and the non-linear terms are linearized by lagging appropriate factors. The modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model. The numerical approximation is stabilized by using an SUPG approximation for the constitutive equation. Under a small data assumption on the true solution, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of the mesh parameter h, the time discretization parameter Δt, and the SUPG coefficient ν are also derived. Numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented.     

Application of lower order integration schemes in linear shell problems

A method is presented by which steady flow solutions may be obtained to problems which involve non-Newtonian memory fluids. The finite element method is used in conjunction with a Galerkin form of the equations of motion and continuity. Integral constitutive laws are directly employed without extra-stress differential equations. The stress is computed by construction of the portion of the streamline lying upstream of element quadrature points. This construction is shown to be quite simple, owing to the special form of finite element trial velocity fields. Two test problems are analyzed which use the integral form of the Maxwell constitutive law. The interaction between the fluid elasticity and solution procedures for the discrete equations is discussed.     

On predicting abrupt contraction flows with differential and algebraic viscoelastic models

The numerical simulation of flows through a planar contraction at low Reynolds number is considered for Newtonian and for viscoelastic fluids. The recently proposed algebraic extra-stress model (AESM) derived from the differential constitutive equation for an Oldroyd-B fluid is extended to a Phan-Thien–Tanner fluid. The approach is based on the exact polynomial representation using a three-term tensor basis. It is also shown that the algebraic formulation reproduces exactly the extra-stress tensor components for pure shear and for pure elongation flow. A parameter based on the strain rate and the rotation rate tensors is presented to identify the regions of the flow where the AESM model produces exact results. A second-order numerical scheme accurate in time and space, based on the finite volume method using a staggered grid has been applied to solve the conservation and constitutive equations for the Newtonian and viscoelastic flows. The numerical simulations for the viscoelastic fluids have been done using the classical constitutive equations in a differential form and the algebraic extra-stress model. Excellent agreement between the extra-stress values is obtained with the two different approaches, showing the viability of AESM.     

Transverse shear stiffness of laminated anisotropic shells

The additional constitutive equations required by transverse shear deformation theory of anisotropic heterogeneous shells are derived without the usual assumption of thickness distribution for either transverse shear stresses or strains. The derivation is based on Taylor series expansions about a generic point for stress resultants and couples which identically satisfy plate equilibrium equations. These equations give the in-surface stress resultants and couples in terms of the transverse shear stress resultants at the point and arbitrary constants, which may be interpreted as redundant “forces”. Starting from these expressions, we derive statically correct expressions (in terms of the transverse shear stress resultants and redundants) of the following variables: (1) in-surface stresses, using the stretching-bending constitutive equations and the Kirchhoff distributions of in-surface strains, (2) transverse shear stresses, by integration in the normal direction of the three-dimensional equilibrium equations, and (3) the area density of transverse shear strain energy, by integration in the normal direction of the corresponding volumetric density. Finally, by applying Castigliano's theorem of least work, the shear strain energy is minimized with respect to the redundants, thereby leading to the desired constitutive equations. Corresponding transverse shear stiffnesses are presented for several laminated walls, and reasonable agreement is obtained between transverse shear deformation plate theory using these stiffnesses and exact three-dimensional elasticity solutions for the problem of cylindrical bending of a plate.     

Aerodynamic shape optimization of supersonic aircraft configurations via an adjoint formulation on distributed memory parallel computers

This work describes the application of a control theory-based aerodynamic shape optimization method to the problem of supersonic aircraft design. A high fidelity computational fluid dynamics (CFD) algorithm modelling the Euler equations is used to calculate the aerodynamic properties of complex three-dimensional aircraft configurations. The design process is greatly accelerated through the use of both control theory and parallel computing. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is then implemented in parallel using a domain decomposition approach, an optimized communication schedule, and the Message Passing Interface (MPI) Standard for portability and efficiency. In our earlier studies, the serial implementation of this design method, was shown to be effective for the optimization of airfoils, wings, wing–bodies, and complex aircraft configurations using both the potential equation and the Euler equations. In this work, our concern will be to extend the methodologies such that the combined capabilities of these new technologies can be used routinely and efficiently in an industrial design environment. The aerodynamic optimization of a supersonic transport configuration is presented as a demonstration test case of the capability. A particular difficulty of this test case is posed by the close coupling of the propulsion/airframe integration.     

A perioral dynamic model for investigating human speech articulation

A multibody system called the perioral dynamic model for investigating dynamic behavior of human speech articulator is presented. The model is based on the biomechanical architecture of human speech articulators, consisting of the soft tissue around the lips, the related muscles, and the jaw bone structure. The dynamic consequence of human speech articulation is revealed as a sequential perioral motion induced by the selectively activated muscle actions. As an anatomically consistent biomechanical platform, the perioral dynamic model is designed to represent the rigid jaw motion as well as the perioral soft tissue deformation interacting with each other. The perioral soft tissue in the model is approximated as a discrete particle system consisting of lumped point masses interconnecting with adjacent ones via viscoelastic elements. To ensure continuum-compatible static deformation in the discrete particle system, we introduce a method of adjusting element stiffness. We also present a new method for determining the effective forces acting on the jaw-bone-attached nodes that transforms jaw dynamics in the rigid body system into the one defined in the discrete particle system, keeping dynamic equivalency and equipollency between two systems. To derive muscle activations which let the developed dynamic model produce a simulated perioral motion mimicking an actual human speech behavior, we present an inverse dynamics technique driven by visual observation-based feedback of an actual lip motion.     

On Computational Modeling in Tumor Growth

The paper addresses modeling of avascular and vascular tumor growth within the framework of continuum mechanics and the adopted numerical solution strategies. The models involve tumor cells, both viable and necrotic, healthy cells, extracellular matrix (ECM), interstitial fluid, neovasculature and co-opted blood vessels, nutrients, waste products, and their interaction and evolution. Attention is focused on the more recent models which are much richer than earlier ones, i.e. they address more aspects of this complicated problem. An important element is how the governing equations are obtained and how the many interfaces between the above listed components are dealt with. These considerations suggest the definition of different classes of models comprised of diffusion, single phase flow and multiphase flow models with or without a solid phase. A multiphase flow model in a deforming porous medium (ECM) is chosen as reference model since it appears to invoke the least number of simplifying assumptions and has the largest potential for further development. The strategies adopted in the choice of the many model dependent constitutive relationships are discussed in detail. Two applications referring to two different model classes conclude the paper.     

Hysteretic upscaled constitutive relationships for vertically integrated porous media flow

Subsurface two-phase flow in porous media often takes place in reservoirs with a high ratio between the associated lateral and vertical extent and the lateral and vertical flow time scales. This allows for a two-scale approach with effective quantities for two-dimensional horizontal flow equations obtained from reconstructed hydrostatic vertical pressure and saturation distributions. Here, we derive explicit expressions for the two dimensional constitutive relationships for a play-type hysteretic Brooks–Corey capillary pressure function with a pore-size distribution index of 2 and quadratic relative permeabilities. We obtain an explicit hysteretic parametrization for the upscaled capillary pressure function and the upscaled relative permeabilities. The size of the hysteresis loop depends on the ratio between buoyancy and the entry pressure, i.e. it scales with the reservoir height and the ratio between drainage and imbibition capillary pressure. We find that the scaling for the relative permeability is non-monotonic and hysteresis vanishes for both small and large reservoirs.     

The optimal solution for the flow of a fourth-grade fluid with partial slip

The steady flow of a non-Newtonian fluid when slippage between the plate and the fluid occurs is considered. The constitutive equations of the fluid are modeled for a fourth-grade non-Newtonian fluid with partial slip; they give rise to nonlinear boundary value problems. Analytical solutions are obtained using powerful analytic techniques for solving nonlinear problems, homotopy perturbation and optimal homotopy asymptotic methods. The results obtained are compared with the numerical results and it is shown that solutions exist for all values of the non-Newtonian parameters. The solutions valid for the no-slip condition for all values of the non-Newtonian parameters can be derived as special cases of the present analysis. Finally the solutions are discussed using a graphical approach.     

Defect correction method for time-dependent viscoelastic fluid flow

A defect correction method for solving the time-dependent viscoelastic fluid flow, aiming at high Weissenberg numbers, is presented. In the defect step, the constitutive equation is computed with the artificially reduced Weissenberg parameter for stability, and the residual is considered in the correction step. We show the convergence of the method and derive an error estimate. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.     

Output pressure and efficiency of electrokinetic pumping of non-Newtonian fluids

Theoretical expressions of the flow rate, output pressure and thermodynamic efficiency of electrokinetic pumping of non-Newtonian fluids through cylindrical and slit microchannels are reported. Calculations are carried out in the framework of continuum fluid mechanics. The constitutive model of Ostwald-de Waele (power law) is used to express the fluid shear stress in terms of the velocity gradient. The resulting equations of flow rate and electric current are nonlinear functions of the electric potential and pressure gradients. The fact that the microstructure of non-Newtonian fluids is altered at solid–liquid interfaces is taken into account. In the case of fluids with wall depletion, both the output pressure and efficiency are found to be several times higher than that obtained with simple electrolytes under the same experimental conditions. Apart from potential applications in electrokinetic pumps, these predictions are of interest for the design of microfluidic devices that manipulate non-Newtonian fluids such as polymer solutions and colloidal suspensions. From a more fundamental point of view, the paper discusses a relevant example of nonlinear electrokinetics.     

Computer implementation aspects for fluid-structure interaction problems

In this paper the computer implementation aspects for fluid-structure interaction problems are presented. Special attention is placed on finite element fluid modelling, implicit-explicit transient algorithms and finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. All these methodologies have been integrated into a working finite element computer code.