Low-order Formulation Research Articles

One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic SN setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

The Effects of Rotation and Solidity on the Aerodynamic Behavior of Low-Pressure Axial Flow Fans

Abstract Implicit axial flow fan models are commonly used for the numerical analysis of industrial heat exchanger systems. However, these fan models perform poorly within the often complex off-design flow environments characteristic of these systems, as their low-order formulations lose applicability under highly 3D flow conditions. At off-design flow conditions, rotation and blade solidity effects cause physical fan blade behavior to deviate from the anticipated 2D behavior characterized by the implicit models. Physical fan blade behavior at off-design flow conditions, however, remains uncertain and the specific effects of rotation and solidity are not yet widely understood. This study investigates off-design fan blade behavior by presenting explicit 3D computations of two low-pressure axial flow fans. The effects of rotation and blade solidity are then identified through a side-by-side comparison of the computed 3D fan blade data against isolated 2D airfoil data. The rotating fan blade lift characteristics are shown to be distinct from the comparative 2D airfoil trends. At low-flow (off-design) operating conditions, rotation establishes large radial flow movements, and the high blade solidities cause standing vortices to develop within the rotating blade passages. The radial outflow energizes the blades' boundary layers, and the vortices inhibit the flow from separating from the blades' surfaces. Consequently, blade stall is avoided and high lift coefficient values are realized. The presented fan blade lift characteristics are unique relative to literature on other turbomachines of lower blade solidity, highlighting the importance of considering solidity effects in implicit rotor model developments.

Leading-edge-vortex tailoring on unsteady airfoils using an inverse aerodynamic approach

In this paper, we present an approach to obtain a desired leading-edge vortex (LEV) shedding pattern from unsteady airfoils through the execution of suitable motion kinematics. Previous research revealed that LEV shedding is associated with the leading-edge suction parameter (LESP) exceeding a maximum threshold. A low-order method called LESP-modulated discrete vortex method (LDVM) was also developed to predict the onset and termination of LEV shedding from an airfoil undergoing prescribed motion kinematics. In the current work, we present an inverse-aerodynamic formulation based on the LDVM to generate the appropriate motion kinematics to achieve a prescribed LESP variation, and thus, the desired LEV shedding characteristics from the airfoil. The algorithm identifies the kinematic state of the airfoil required to attain the target LESP value through an iterative procedure performed inside the LDVM simulation at each time step. Several case studies are presented to demonstrate design scenarios such as tailoring the duration and intensity of LEV shedding, inducing LEV shedding from the chosen surface of the airfoil, promoting or suppressing LEV shedding during an unsteady motion on demand, and achieving similar LEV shedding patterns using different maneuvers. The kinematic profiles generated by the low-order formulation are also simulated using a high-fidelity unsteady Reynolds-averaged Navier–Stokes method to confirm the accuracy of the low-order model.

Effect of Clocking on Compressor Noise Generation

The effect of stator clocking on the acoustic noise-generation characteristics in an axial high-pressure compressor is analyzed. A realistic geometry with 1.5 stages is assessed using high-fidelity and low-order numerical methods for different clocking positions at approach operating conditions. The compressor efficiency and the acoustic noise emission are found to vary insignificantly between the simulated clocking configurations. Nonetheless, the pressure distribution is altered significantly, right upstream of the inlet guide vanes. Although the cut-on modes exhibit at least 10-dB-higher amplitudes, the cutoff modes contribute decisively to the wave pattern in the near field. An optimal acoustic liner design can expand on the differently evolving interference pattern of acoustic waves at discrete frequencies. The low-order model is found to predict the directionality of the acoustic waves, and the cut-on criteria for the individual modes is in excellent agreement with the high-fidelity simulations. However, the phase cannot be estimated due to the simplicity of the low-order formulation.

A low order 3D virtual element formulation for finite elasto–plastic deformations

This work addresses an efficient low order 3D virtual element method for elastic–plastic solids undergoing large deformations. Virtual elements were introduced in the last decade and applied to various problems in solid mechanics. The successful application of the method to non-linear problems such as finite strain elasticity and plasticity in 2D leads naturally to the question of its effectiveness and robustness in the third dimension. This work is concerned with the extensions of the virtual element method to problems of 3D finite strain plasticity. Low-order formulations for problems in three dimensions, with elements being arbitrary shaped polyhedra, are considered. The formulation is based on minimization of a pseudo energy expression, with a generalization of a stabilization techniques, introduced for two dimensional polygons, to the three-dimensional domain. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties. For comparison purposes, results of the standard finite element method are also demonstrated.

A new incompatible mode element with selective mass scaling for saturated soil dynamics

It is a well-known fact that addressing hydromechanical problems in saturated soils with the finite element method and equal-order interpolation formulations in displacements and pore pressures produces unstable results. Classically, stabilization has been achieved by increasing the interpolation degree of displacement with respect to pore pressure, hence fulfilling the Babuska–Brezzi condition. However, the use of quadratic elements involves high computational costs. From that point of view, the use of stabilized low-order elements is a more desirable option. Much research has been carried out in different directions in the stabilization of low-order formulations for saturated soils in quasistatic conditions, among others with the technique based on strain field enhancement through internal degrees of freedom. This article presents an alternative displacement–pore pressure formulation for saturated soil dynamics based on the enhancement of the displacement field through incompatible modes.

Modelling the generation of temperature inhomogeneities by a premixed flame

The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is investigated and modelled, with emphasis on the generation of ‘entropy waves’, i.e. entropic inhomogeneities of downstream temperature. Transient computational fluid dynamics simulations of two adiabatic lean methane-air flames of different Péclet numbers provide guidance and validation data for subsequent modelling. The respective entropy transfer functions, which describe the production of temperature inhomogeneities, as well as transfer functions for the variation of the heat release, are determined from the computational fluid dynamics time series data by means of system identification. The processes governing the dynamics of the entropy transfer functions are segregated into two sub-problems: (1) heat release due to chemical reaction at the flame front and (2) advective and diffusive transport. By adopting a formulation in terms of a mixture fraction variable, these two sub-problems can be treated independently from each other. Models for both phenomena are derived and analysed using simple 0- and 1-dimensional configurations. The heat release process (1) is represented by a fast-reaction-zone model, which takes into account variations of the specific heat capacity with equivalence ratio in order to evaluate the magnitude of downstream temperature fluctuations with quantitative accuracy. For the transport processes (2), two types of models based on mean field data from the computational fluid dynamics simulation are proposed: A semi-analytical, low-order formulation based on stream lines, and a state-space formulation, which is constructed by Finite Elements discretisation of the transport equation for mixture fraction. Model predictions for the entropy transfer functions are found to agree well with the computational fluid dynamics reference data at very low computational costs.

A low order virtual element formulation for finite elasto-plastic deformations

The virtual element method has been developed over the last decade and applied to problems in elasticity and other areas. The successful application of the method to linear problems leads naturally to the question of its effectiveness in the nonlinear regime. This work is concerned with extensions of the virtual element method to problems of finite strain plasticity. Low-order formulations for problems in two dimensions, with elements being arbitrary polygons, are considered. The formulation is based on minimization of an incremental energy expression, with a novel construction of the stabilization energy for elasto-plasticity. The resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency, accuracy and convergence properties.

Efficient virtual element formulations for compressible and incompressible finite deformations

The virtual element method has been developed over the last decade and applied to problems in elasticity and other areas. The successful application of the method to linear problems leads naturally to the question of its effectiveness in the nonlinear regime. This work is concerned with extensions of the virtual element method to problems of compressible and incompressible nonlinear elasticity. Low-order formulations for problems in two dimensions, with elements being arbitrary polygons, are considered: for these, the ansatz functions are linear along element edges. The various formulations considered are based on minimization of energy, with a novel construction of the stabilization energy. The formulations are investigated through a series of numerical examples, which demonstrate their efficiency, convergence properties, and for the case of nearly incompressible and incompressible materials, locking-free behaviour.

CFD-based application of the Nyquist criterion to thermo-acoustic instabilities

A novel approach for the analysis of self-excited instabilities in thermo-acoustic systems is proposed. Combining computational fluid dynamics with low-order acoustic modeling, the open-loop transfer function of the system under investigation is computed. The system eigenmodes and the linear stability characteristics are then deduced from a Nyquist plot. The method is suitable for systems where – due to geometrical complexities or non-compact regions of heat release – a low-order formulation is not appropriate or not available. Explicit knowledge of the frequency response or the transfer matrix of the heat source is not required. Further advantages of the new approach are discussed in the paper. To establish proof of concept, the method is validated against a simple model of a Rijke tube. Over the frequency range considered, frequencies and growth rates of stable as well as unstable eigenmodes are predicted accurately. The new method, combining flow simulation, low-order acoustic modeling and control theory, makes possible a comprehensive analysis of acoustic stability behavior of complex thermo-acoustic systems at comparatively modest computational cost.

A homogenized free energy model for hysteresis in thin-film shape memory alloys

Thin-film shape memory alloys (SMAs) have become excellent candidates for microactuator fabrication in microelectromechanical systems due to their capability to achieve very high work densities, produce large deformations, and generate high stresses. In general, the material behavior of SMAs is nonlinear and hysteretic. To achieve the full potential of SMA actuators, it is necessary to develop models that characterize the nonlinearities and hysteresis inherent to the constituent materials. We develop a model that quantifies the nonlinearities and hysteresis inherent to SMAs. The fully thermomechanical model is based on free energy principles combined with stochastic homogenization techniques. It predicts rate-dependent, polycrystalline SMA behavior, and it accommodates heat transfer issues pertinent to thin-film SMAs. The main advantages of this model are that it admits a simple, low-order formulation suitable for implementation and subsequent control design, and that most of the model parameters are identifiable directly from standard measurements. We illustrate aspects of the model through comparison with thin-film SMA superelastic and shape memory effect hysteresis data.

A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I-geometrically linear applications

Accuracy and efficiency are the main features expected in finite element method. In the field of low-order formulations, the treatment of locking phenomena is crucial to prevent poor results. For three-dimensional analysis, the development of efficient and accurate eight-node solid-shell finite elements has been the principal goal of a number of recent published works. When modelling thin- and thick-walled applications, the well-known transverse shear and volumetric locking phenomena should be conveniently circumvented. In this work, the enhanced assumed strain method and a reduced in-plane integration scheme are combined to produce a new eight-node solid-shell element, accommodating the use of any number of integration points along thickness direction. Furthermore, a physical stabilization procedure is employed in order to correct the element's rank deficiency. Several factors contribute to the high computational efficiency of the formulation, namely: (i) the use of only one internal variable per element for the enhanced part of the strain field; (ii) the reduced integration scheme; (iii) the prevention of using multiple elements' layers along thickness, which can be simply replaced by any number of integration points within a single element layer. Implementation guidelines and numerical results confirm the robustness and efficiency of the proposed approach when compared to conventional elements well-established in the literature. Copyright © 2004 John Wiley & Sons, Ltd.

Thermal convection in a cylindrical annulus heated laterally

In this paper we study thermoconvective instabilities appearing in a fluid within a cylindrical annulus heated laterally. As soon as a horizontal temperature gradient is applied a convective state appears. As the temperature gradient reaches a critical value a stationary or oscillatory bifurcation may take place. The problem is modelled with a novel method which extends the one described in (numerico). The Navier Stokes equations are solved in the primitive variable formulation, with appropriate boundary conditions for pressure. This is a low order formulation which in cylindrical coordinates introduces lower order singularities. The problem is discretized with a Chebyshev collocation method easily implemented and its convergence has been checked. The results obtained are not only in very good agreement with those obtained in experiments, but also provide a deeper insight into important physical parameters developing the instability, which has not been reported before.

Low order plate bending elements with enhanced strains

Two variational formulations are considered for the development of low order plate bending elements. For the constitutive equation, the three-dimensional stress–strain relations are used without any modification. No shear correction factors are necessary in the formulation. Bilinear functions are used to approximate the deflection, the two rotations and the thickness change of the plate. The key ingredients in the low order formulation are the inclusion of the thickness change and the use of enhanced strain terms.

Application of the Taylor Polycrystal Plasticity Model to Complex Deformation Experiments

The extended Taylor assumption of uniform deformation gradient among grains was applied in 3-D polycrystal plasticity simulations for complex loading paths at finite strain for OFHC Cu using the Los Alamos polycrystal plasticity (LApp) code (Kocks et al., 1994). Comparisons of both stress-strain behavior and texture evolution, with and without the inclusion of latent hardening effects, show that the theory overpredicts the rate of development of texture in both torsion and compression. Compression stress-strain behavior was accurately predicted, but the effect of the prestrain, either compressive or torsional, on subsequent nonproportional deformation response was inadequately modeled. Some possible sources of the discrepancies are discussed, including the low order nature of the extended Taylor model for intergranular interactions as compared to self-consistent models, low order formulation of slip system hardening, lack of accounting for formation of dislocation substructure within grains, and the possible role of anisotropic elasticity. Deformation-induced anisotropy and accommodation of intergranular constraint afforded by geometrically necessary dislocation substructure formation is viewed as the key neglected element of the formulation.