Generalized Petrov-Galerkin time finite element weighted residual methodology for designing high-order unconditionally stable algorithms with controllable numerical dissipation
Introduction
Temporal discretized operators are fundamental and crucial for transient simulations in terms of predicting the physics in engineering applications, such as fluid flow, heat transfer and structural dynamics. In general, unconditionally stable algorithms are preferred as they can use a large time step to obtain stable results with a significant payoff regarding the computational cost; whereas, in the case with large stiffness, high-frequency components lead to physically incorrect and/or numerically oscillatory results and may blow up the simulations. To eliminate those nonphysical and/or numerical oscillations, the time discretization algorithms are usually expected to have controllable numerical dissipation. Different numerical approximations, encompassing finite difference method [1], [2], [3], [4], generalized time-weighted residual method [5], [6], [7], finite element formulations in space and time [8], [9], [10], and the like, have been utilized to deriving computational frameworks for time discretization, such as linear multi-step methods, θ-method (generalized Trapezoidal family of algorithms, or generalized-α method for first-order systems), and Runge-Kutta-type methods, etc. It has been pointed out [11], [12], [13], [14], [15], [16], [17], [18] that most of these algorithms can be recast in a unified manner, that is to derive them from a generalized time-weighted residual methodology. Inspired by this generalization concept, this exposition starts from the generalized time-weighted procedure, while integrating with Petrov-Galerkin time finite element method, to propose a unified computational framework. That is, we can develop and design high-order unconditionally stable techniques for temporal discretization, and additionally with controllable numerical oscillation behavior in high frequency.
Regarding existing first-order time accurate algorithms, the unconditionally stable Euler backward algorithm is the only L-stable method in the generalized Trapezoidal family, showing excellent dissipation features and yielding smooth results with a large time step Δt (in high frequency). However, the Euler backward algorithm is limited by its low order of accuracy and may be insufficient for practical applications. The only second-order algorithm in the generalized Trapezoidal family is the Crank-Nicolson method, which has no dissipation and shows oscillatory behavior in the high frequency. In terms of second-order time accurate algorithms with controllable numerical dissipation, the Generalized Single-Step Single-Solve family of algorithms for transient first-order systems (GS4-1) and second-order systems (GS4-2) developed by Prof. Tamma's research group [11], [12], [13], [14], [15], [16], [17], [18] show a general representation encompassing the entire class of LMS methods involving a single-step and a single-solve representation within each time step. In GS4-1, two free parameters named principle root and spurious root are introduced to control the numerical damping in high frequency. Specifically, recovers the Crank-Nicolson method without numerical dissipation; recovers Gear's method with the greatest numerical damping; generates new algorithms with selectively controllable numerical dissipation. However, we cannot get higher-order (greater than 2) A/L stable algorithms with this GS4-1 structure according to Dahlquist first barrier theorem [19] (also see the generalized barrier theorem by Zhou and Tamma (Ref. [5])).
A possible technique to formulate high-order time integration algorithms is introducing the concept of Bubnov Galerkin theory to approximate the time dimension. Similar to that in space finite element methods, high-order basis function usually leads to high-order numerical accuracy. The time weighted residual method was initially proposed by Zienkiewicz and Taylor in [4] for the Single-step p,j method and Generalized Newmark p,j methods. The specific pros/cons regarding these algorithms were carefully investigated and advanced more generally to the entire GS4-1/-2 family of algorithms in [5], [20]. Regarding applying the Petrov-Galerkin discretization in time, several pioneering studies have been reported in Refs. [21], [22], [23], [24], [25] in which -order polynomials were employed for the solution and th for the test functions and the applications covered both first-order systems (heat transfer problems) and second-order systems (wave propagation problems). Moreover, various time finite element methods can be found in [26], [27], [28], [29] and the references therein. Due to the diversity of the Galerkin methodology, the concept of discontinuous Galerkin [30], [31], [32], [33] is also introduced for improved physics and computational performance under certain situations. Moreover, it is worth noting that some of the time finite element methods are embedded in the framework of space-time system [34], [35], [36], [37], [38], that is to say, both space and time dimensions are consistently discretized and approximated in the manner of Galerkin theory. These classical space-time methods are designed at a system level; in particular, the n-dimensional problem in space is ultimately escalated to -dimensional problem, and as such computational efficiency is a drawback inherent to the space-time finite element method. Besides, spectral element methods have also been extended to this space-time element concept [38], [39], [40], [41].
In this paper, we aim to develop a unified framework to design high-order A/L-stable algorithms in the realm of the stabilized time-weighted method. To this end, we apply the fundamental Petrov-Galerkin theory to the time dimension that inherently provides us a wide design range of algorithms. All the desired algorithmic properties (‘wish list’): unconditionally stable, pth-order accuracy, and controllable numerical dissipation, are imposed as constraint equations to the free algorithmic parameters via the concept of Algorithms by Design [5], [6], [7], [42], [43], leading to a unified framework to design advanced time finite element framework. Quadratic and cubic basis functions are utilized as illustrative examples to present design details for third-/fifth-order unconditionally stable algorithms with controllable numerical dissipation features; moreover, second-/fourth-order algorithms can be designed with different accuracy constraints as well. In addition, this generalized Petrov-Galerkin time finite element framework can also permit other basis functions for higher-order accurate algorithms with controllable numerical dissipation, and can be readily integrated with higher-order spatial discretization techniques to formulate space-time methodologies with arbitrarily high orders of accuracy in space and time.
Section snippets
Petrov-Galerkin time-weighted residual methodology
Consider the following general semi-discretized linear first-order transient system with initial conditions: where denotes the first derivative of u upon time t. C and K are system matrices generated by space discretization techniques, such as finite/spectral element method (FEM/SEM), finite volume method (FVM), finite difference method (FDM), isogeometric analysis, Discontinuous Galerkin method (DGM), and the like. Firstly, we cast Eq. (1) into a generalized
Design of algorithm structure
Considering a homogeneous problem with single degree of freedom, . Applying the quadratic basis functions to design algorithms under the umbrella of the Petrov-Galerkin time-weighted residual framework, we have the following design details:
Step 1 As illustrated in Fig. 1, choose three points, , as the interpolation points. The time increment is firstly mapped to an isoparametric element of by:
Step 2 The generalized quadratic basis functions are
Design example-II: Cubic basis functions leading to fifth-order accurate algorithms with controllable numerical dissipation
In this section, we give a brief design procedure via using cubic basis function and show the parameters for fifth-order accurate algorithms with controllable numerical dissipation. As illustrated in Fig. 3, choose four points, , as the interpolation points, and therefore the generalized cubic basis functions are defined as:
Single degree of freedom problem
In this section, we first consider the following single degree of freedom (SDOF) problem: where the exact solution is given by: and the numerical error is defined as:
For the family of algorithms, Fig. 5 presents the convergence rate plots for different selections of . It can be found that all algorithms with different selections of show third-order
Conclusions
In this paper, a stabilized Petrov-Galerkin time finite element weighted residual methodology was proposed to preserve unconditional stability, high-order accuracy, and features with controllable numerical dissipation for the time discretization. Different from existing time finite element methods, the proposed unified computational framework introduced a stabilized term that achieves controllable numerical dissipation of high-frequency components. Quadratic and cubic basis functions were
CRediT authorship contribution statement
Yazhou Wang: Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Kumar K. Tamma: Conceptualization, Methodology, Project administration, Supervision, Writing – original draft, Writing – review & editing. Tao Xue: Formal analysis, Methodology, Validation. Dean Maxam: Formal analysis, Methodology, Validation. Guoliang Qin: Funding acquisition, Project administration, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 51776155). The author Yazhou Wang would like to thank for the financial support from China Scholarship Council (No. 201906280340). Thanks are also due to Prof. Tamma's computational mechanics laboratory at the University of Minnesota.
References (47)
- et al.
Generalized heat conduction model involving imperfect thermal contact surface: application of the GSSSS-1 differential-algebraic equation time integration
Int. J. Heat Mass Transf.
(2018) - et al.
A two-field state-based peridynamic theory for thermal contact problems
J. Comput. Phys.
(2018) - et al.
A consistent moving particle system simulation method: applications to parabolic/hyperbolic heat conduction type problems
Int. J. Heat Mass Transf.
(2016) - et al.
On the treatment of high-frequency issues in numerical simulation for dynamic systems by model order reduction via the proper orthogonal decomposition
Comput. Methods Appl. Mech. Eng.
(2017) A space-time finite element method for the wave equation
Comput. Methods Appl. Mech. Eng.
(1993)- et al.
A space-time discontinuous Galerkin method for the incompressible Navier–Stokes equations
J. Comput. Phys.
(2013) - et al.
Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations
J. Comput. Phys.
(2006) - et al.
Space-time discontinuous Galerkin method for advection–diffusion problems on time-dependent domains
Appl. Numer. Math.
(2006) - et al.
A space-time discontinuous Galerkin method for scalar conservation laws
Comput. Methods Appl. Mech. Eng.
(2004) - et al.
Space-time finite element methods for second-order hyperbolic equations
Comput. Methods Appl. Mech. Eng.
(1990)