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Lee, Seunggyu; Yoon, Sungha; Kim, Junseok
doi: 10.1002/nme.7288pmid: N/A
In this article, we present an unconditionally energy stable linear scheme for the Cahn–Hilliard equation with a high‐order polynomial free energy. The classical Cahn–Hilliard equation does not satisfy the maximum principle; hence the order parameter can be shifted out of the minimum values of the double‐well potential. We adopt a high‐order polynomial potential to diminish this effect and employ the efficient linear convex splitting scheme. Since the stabilizing factor gradually increases as the degree of potential becomes greater, we modify a non‐physical part of potential as a fourth‐order polynomial to reduce the stabilizing factor. Numerical results as well as theoretical results demonstrate the accuracy and energy stability of our method. Furthermore, we verify that some limitations arising from applications of the classical Cahn–Hilliard model can be resolved by adopting a high‐order free energy.
Xiao, B; Natarajan, S; Birk, C; Ooi, EH; Song, C; Ooi, ET
doi: 10.1002/nme.7287pmid: N/A
A general technique to develop arbitrary‐sided polygonal elements based on the scaled boundary finite element method is presented. Shape functions are derived from the solution of the Poisson's equation in contrast to the well‐known Laplace shape functions that are only linearly complete. The application of the Poisson shape functions can be complete up to any specific order. The shape functions retain the advantage of the scaled boundary finite element method allowing direct formulation on polygons with arbitrary number of sides and quadtree meshes. The resulting formulation is similar to the finite element method where each field variable is interpolated by the same set of shape functions in parametric space and differs only in the integration of the stiffness and mass matrices. Well‐established finite element procedures can be applied with the developed shape functions, to solve a variety of engineering problems including, for example, coupled field problems, phase field fracture, and addressing volumetric locking in the near‐incompressibility limit by adopting a mixed formulation. Application of the formulation is demonstrated in several engineering problems. Optimal convergence rates are observed.
Lehrenfeld, Christoph; Stocker, Paul
doi: 10.1002/nme.7258pmid: N/A
In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz‐type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non‐constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.
Dobrilla, Simona; Matthies, Hermann G.; Ibrahimbegovic, Adnan
doi: 10.1002/nme.7289pmid: N/A
This work tackles the issue of identifiability of fracture and bond properties in reinforced concrete. The basis for modeling of fracture is a computational model capable of describing damage and failure mechanisms in concrete, as well as bond‐slip which is a result of degradation of the concrete‐steel interface. The discrete approximation combines ED‐FEM for concrete crack representation in each element and X‐FEM representation of bond‐slip along a particular reinforcement bar. The uncertain model parameters are modeled as random variables and identified via Bayesian inference with the help of observations from tensile tests on concrete tie beams with a single embedded reinforcement bar. We discuss how the choice of observation type affects the parameter identifiability and propose combinations which improve the estimation capabilities and reduce the discrepancy between the computed and observed quantities of interest.
doi: 10.1002/nme.7297pmid: N/A
Peridynamics (PD) is a nonlocal continuum theory capable of handling fracture mechanisms with ease. However, its use involves high computational costs. On the other hand, Carrera Unified Formulation (CUF) allows one to use one‐dimensional high‐order finite elements, resulting in excellent accuracy while improving computational efficiency. To address the high computational cost of solving fracture problems, a coupling technique between these two theories is necessary. Various approaches have been proposed to couple peridynamic grids with finite element meshes in the literature. However, most of these approaches are affected by arbitrary choices of blending functions and tuning parameters or exhibit spurious effects at the interfaces. To overcome these issues, we propose a simple coupling technique based on overlapping PD/CUF regions and continuity of the displacement field at the interfaces. This approach is verified through static analysis of classical beams and thin‐walled structures with applications in the aerospace industry.
Sevilla, Ruben; Duretz, Thibault
doi: 10.1002/nme.7294pmid: N/A
A new face‐centred finite volume method (FCFV) for Stokes problems involving sharp interfaces is proposed. Two formulations, based on two strong forms of the Stokes problem, and using different mixed variables, are presented. Particular attention is paid to the symmetry of the resulting system of global equations, and a simple rewriting of the interface boundary condition is proposed to ensure that one of the formulations preserves the symmetry of the linear system that is usually lost when considering material interfaces. Four numerical examples are considered to test the implementation numerically by performing mesh convergence studies, in two and three dimensions. The examples account for discontinuous viscosity as well as the effect of surface tension. The results show that one of the formulations is less sensitive to the numerical stabilisation used in FCFV methods but does not preserve the symmetry of the global system, whereas the other formulation is more sensitive to the stabilisation, but preserves the symmetry of the resulting system of equations. The FCFV method appears as a promising alternative for the simulation of viscous flow involving internal boundaries on conformal meshes. The potential application of the FCFV method for the purpose of geodynamic modelling is discussed.
Li, Jinze; Yu, Kaiping; Zhao, Rui; Fang, Yong
doi: 10.1002/nme.7291pmid: N/A
This paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters (ρ2∞$$ {\rho}_2^{\infty } $$ and ρ3∞$$ {\rho}_3^{\infty } $$) to control numerical dissipation imposed in the second and third sub‐steps. The parameter ρ3∞$$ {\rho}_3^{\infty } $$ adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter ρ2∞$$ {\rho}_2^{\infty } $$ can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via ρ2∞$$ {\rho}_2^{\infty } $$. The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.
Chen, Hao; Zhao, Haibo; Huang, Tao; Yu, Peng; Shu, Chang
doi: 10.1002/nme.7290pmid: N/A
A unified numerical model for simulating thermal flows past and through porous media whose thermal properties are identical with those of the surrounding fluid is proposed in this paper, which advances the work of the unified immersed boundary‐lattice Boltzmann flux solver (UIB‐LBFS). The proposed method greatly simplifies the conventional computation procedure for thermal flows past porous bodies by unifying the governing equations for the flow and temperature fields in both the porous and fluid domains. This is achieved by introducing a diffuse layer, across which the variation of the flow properties including its thermal features are expected to be smooth. Also, the developed method can be applied on either uniform or nonuniform grid since the numerical fluxes are locally reconstructed by LBFS at cell interfaces. In addition, the boundary conditions for both the flow and temperature fields at the porous‐fluid interface with complex shape can be implemented through IBM in a simple way. The proposed method is validated by several numerical examples over wide ranges of thermal parameters from forced convection to mixed convection. Excellent agreements are obtained for all simulations, including the one with severe temperature gradient in the thin temperature boundary layer at Pr=10$$ \mathit{\Pr}=10 $$, which demonstrates the capabilities of the proposed method.
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