The purpose of this paper is to describe a novel implementation of a multispatial method, multitime-scheme subdomain differential algebraic equation (DAE) framework allowing a mix…
Abstract
Purpose
The purpose of this paper is to describe a novel implementation of a multispatial method, multitime-scheme subdomain differential algebraic equation (DAE) framework allowing a mix of different space discretization methods and different time schemes by a robust generalized single step single solve (GS4) family of linear multistep (LMS) algorithms on a single body analysis for the first-order nonlinear transient systems.
Design/methodology/approach
This proposed method allows the coupling of different numerical methods, such as the finite element method and particle methods, and different implicit and/or explicit algorithms in each subdomain into a single analysis with the GS4 framework. The DAE, which constrains both space and time in multi-subdomain analysis, combined with the GS4 framework ensures the second-order time accuracy in all primary variables and Lagrange multiplier. With the appropriate GS4 parameters, the algorithmic temperature rate variable shift can be matched for all time steps using the DAE. The proposed method is used to solve various combinations of spatial methods and time schemes between subdomains in a single analysis of nonlinear first-order system problems.
Findings
The proposed method is capable of coupling different spatial methods for multiple subdomains and different implicit/explicit time integration schemes in the GS4 framework while sustaining second-order time accuracy.
Originality/value
Traditional approaches do not permit such robust and flexible coupling features. The proposed framework encompasses most of the LMS methods that are second-order time accurate and unconditionally stable.
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Tianhong Ouyang and Kumar K. Tamma
Thermal solidification processes are an important concern in today’smanufacturing technology. Because of the complex geometric nature ofreal‐world problems, analytical techniques…
Abstract
Thermal solidification processes are an important concern in today’s manufacturing technology. Because of the complex geometric nature of real‐world problems, analytical techniques with closed‐form solutions are scarce and/or not feasible. As a consequence, various numerical techniques have been employed for the numerical simulations. Of interest in the present paper are thermal solidification problems involving single or multiple arbitary phases. In order to effectively handle such problems, the finite element method is employed in conjunction with adaptive time stepping approaches to accurately and effectively track the various phase fronts and describe the physics of phase front interactions and thermal behaviour. In conjunction with the enthalpy method which is employed to handle the latent heat release, a fixed‐grid finite element technique and an automatic time stepping approach which uses the norm of the temperature distribution differences between adjacent time step levels to control the error are employed with the scale of the norm being automatically selected. Several numerical examples, including single and multiple phase change problems, are described.
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The purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order…
Abstract
Purpose
The purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented.
Design/methodology/approach
The GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept.
Findings
The present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency.
Originality/value
The novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations.
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Kumar K. Tamma and Sudhir B. Railkar
The present paper describes the applicability of hybrid transfinite element modelling/analysis formulations for non‐linear heat conduction problems involving phase change. The…
Abstract
The present paper describes the applicability of hybrid transfinite element modelling/analysis formulations for non‐linear heat conduction problems involving phase change. The methodology is based on application of transform approaches and classical Galerkin schemes with finite element formulations to maintain the modelling versatility and numerical features for computational analysis. In addition, in conjunction with the above, the effects due to latent heat are modelled using enthalpy formulations to enable a physically realistic approximation to be effectively dealt computationally for materials exhibiting phase change within a narrow band of temperatures. Pertinent details of the approach and computational scheme adapted are described in technical detail. Numerical test cases of comparative nature are presented to demonstrate the applicability of the proposed formulations for numerical modelling/analysis of non‐linear heat conduction problems involving phase change.
Kumar K. Tamma, Xiangmin Zhou and Desong Sha
The time‐discretization process of transient equation systems is an important concern in computational heat transfer applications. As such, the present paper describes a formal…
Abstract
The time‐discretization process of transient equation systems is an important concern in computational heat transfer applications. As such, the present paper describes a formal basis towards providing the theoretical concepts, evolution and development, and characterization of a wide class of time discretized operators for transient heat transfer computations. Therein, emanating from a common family tree and explained via a generalized time weighted philosophy, the paper addresses the development and evolution of time integral operators [IO], and leading to integration operators [InO] in time encompassing single‐step integration operators [SSInO], multi‐step integration operators [MSInO], and a class of finite element in time integration operators [FETInO] including the relationships and the resulting consequences. Also depicted are those termed as discrete numerically assigned [DNA] algorithmic markers essentially comprising of both: the weighted time fields, and the corresponding conditions imposed upon the dependent variable approximation, to uniquely characterize a wide class of transient algorithms. Thereby, providing a plausible standardized formal ideology when referring to and/or relating time discretized operators applicable to transient heat transfer computations.
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Christianne V.D.R. Anderson and Kumar K. Tamma
We first provide an overview of some predominant theoretical methods currently used for predicting thermal conductivity of thin dielectric films: the equation of radiative…
Abstract
We first provide an overview of some predominant theoretical methods currently used for predicting thermal conductivity of thin dielectric films: the equation of radiative transfer, the temperature‐dependent thermal conductivity theories based on the Callaway model, and the molecular dynamics simulation. This overview also highlights temporal and spatial scale issues by looking at a unified theory that bridges physical issues presented in the Fourier and Cattaneo models. This newly developed unified theory is the so‐called C‐ and F‐processes constitutive model. This model introduces the notion of a new dimensionless heat conduction model number, which is the ratio of the thermal conductivity of the fast heat carrier F‐processes to the total thermal conductivity comprised of both the fast heat carriers F‐processes and the slow heat carriers C‐processes. Illustrative numerical examples for prediction of thermal conductivity in thin films are primarily presented.
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Yazhou Wang, Kumar K. Tamma, Dean J. Maxam and Tao Xue
This paper aims to design and analyze implicit/explicit/semi-implicit schemes and a universal error estimator within the Generalized Single-step Single-Solve computational…
Abstract
Purpose
This paper aims to design and analyze implicit/explicit/semi-implicit schemes and a universal error estimator within the Generalized Single-step Single-Solve computational framework for First-order transient systems (GS4-I), which also fosters the adaptive time-stepping procedure to improve stability, accuracy and efficiency applied for fluid dynamics.
Design/methodology/approach
The newly proposed child-explicit and semi-implicit schemes emanate from the parent implicit GS4-I framework, providing numerous options with flexible and controllable numerical properties to the analyst. A universal error estimator is developed based on the consistent algorithmic variables and it works for all the developed methods. Applications are demonstrated by merging the developed algorithms into the iterated pressure-projection method for incompressible Navier–Stokes equations.
Findings
The child-explicit GS4-I has improved solution accuracy and stability properties, and the most stable option is the child explicit GS4-I(0,0)/second-order backward differentiation formula/Gear’s methods, which is new and novel. Numerical tests validate that the universal error estimator emanating from implicit designs works well for the newly proposed explicit/semi-implicit algorithms. The iterative pressure-correction projection algorithm is efficiently fostered by the error estimator-based adaptive time-stepping.
Originality/value
The implicit/explicit/semi-implicit methods within a unified computational framework are easy to implement and have flexible options in practical applications. In contrast to traditional error estimators, which work only on an algorithm-by-algorithm basis, the proposed error estimator is universal. They work for the entire class of implicit/explicit/semi-implicit linear multi-step methods that are second-order time accurate. Based on the accurately estimated local error, balance amongst stability, accuracy and efficiency can be well achieved in the dynamic simulation.
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Yazhou Wang, Dehong Luo, Xuelin Zhang, Zhitao Wang, Hui Chen, Xiaobo Zhang, Ningning Xie, Shengwei Mei, Xiaodai Xue, Tong Zhang and Kumar K. Tamma
The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF…
Abstract
Purpose
The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.
Design/methodology/approach
The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.
Findings
The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.
Originality/value
In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.
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Yazhou Wang, Ningning Xie, Likun Yin, Tong Zhang, Xuelin Zhang, Shengwei Mei, Xiaodai Xue and Kumar Tamma
The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational…
Abstract
Purpose
The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.
Design/methodology/approach
The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.
Findings
The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.
Originality/value
To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.
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This paper gives a review of the finite element techniques (FE)applied in the area of material processing. The latest trends in metalforming, non‐metal forming and powder…
Abstract
This paper gives a review of the finite element techniques (FE) applied in the area of material processing. The latest trends in metal forming, non‐metal forming and powder metallurgy are briefly discussed. The range of applications of finite elements on the subjects is extremely wide and cannot be presented in a single paper; therefore the aim of the paper is to give FE users only an encyclopaedic view of the different possibilities that exist today in the various fields mentioned above. An appendix included at the end of the paper presents a bibliography on finite element applications in material processing for the last five years, and more than 1100 references are listed.