A finite difference method for a singularly perturbed 2‐D elliptic convection‐diffusion PDEs on Shishkin‐type meshes with non‐smooth convection and source terms

Shiromani, Ram; Shanthi, V.; Vigo‐Aguiar, Jesús

doi:10.1002/mma.8877

Cited by 10 publications

(4 citation statements)

References 46 publications

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“…To ensure the accuracy of the FEA, a mesh convergence study is required to determine the optimal mesh size [31][32][33][34][35]. Figure 2(a) displays the results of the mesh convergence study which was conducted by gradually reducing the mesh size until the desired results with a reasonable computational time was obtained.…”

Section: Mesh Convergence Study and Mesh Typementioning

confidence: 99%

Assessment of Buckling Failure in Oil Storage Tanks: Finite Element Simulation of Combined Internal and External Pressure Scenarios

Mashiyane,

Tartibu,

Salifu

2024

Sci. Eng. Technol.

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Investigating the structural behaviour and buckling susceptibility of cylindrical oil storage tanks is crucial for ensuring their safe and reliable operation. In this study, the structural behaviour and buckling susceptibility of closed roof-top cylindrical oil storage tanks under combined internal and external pressure scenarios were investigated using the finite element analysis (FEA) technique. By utilizing the FEA technique, the combined effect of wind-induced pressure of 250 Pa, applied on the outer surface of the tank and internal pressure of 0.5 MPa, applied on the outer surface of the storage was analysed. The result reveals significant stress concentrations and deformation patterns, particularly on the windward side of the tank, thus, emphasizing the susceptibility of the storage tank to buckling under the specified operating conditions for both the filled and half-filled tank; with internal pressure emerging as the primary contributor to mechanical strain and deformation experienced in the tank, while the wind load plays a secondary but significant role in the deformation of the tank. The fe-safe predicted useful life shows that under the specified operating conditions, the filled storage tank will survive 1 429 hours (2 months) while the half-filled tank will survive 3 551 hours (5 months) before failure due to buckling. Thus, the useful life estimation results show the importance of varying oil levels and operating conditions in the structural assessments of storage tanks.

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Section: Mesh Convergence Study and Mesh Typementioning

confidence: 99%

Assessment of Buckling Failure in Oil Storage Tanks: Finite Element Simulation of Combined Internal and External Pressure Scenarios

Mashiyane,

Tartibu,

Salifu

2024

Sci. Eng. Technol.

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“…This is due to the fact that the two‐dimensional layers are moving in time. The mesh movements in time are very difficult to detect when one wants to use a priori‐defined mesh 41 . In the present case, we have generated the mesh by solving moving mesh PDEs in (A6).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media

Kumar,

Das,

Kumar

2024

Numerical Methods in Fluids

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In this work, we consider the Darcy scale precipitation–dissolution reactive transport 1D and 2D models in a porous medium and provide the adaptive mesh based numerical approximations for solving them efficiently. These models consist of a convection‐diffusion‐reaction PDE with reactions being described by an ODE having a nonlinear, discontinuous, possibly multi‐valued right hand side describing precipitate concentration. The bulk concentration in the aqueous phase develops fronts and the precipitate concentration is described by a free and time‐dependent moving boundary. The time adaptive moving mesh strategy, based on equidistribution principle in space and governed by a moving mesh PDE, is utilized and modified in the context of present problem for finite difference set up in 1D and finite element set up in 2D. Moreover, we use a predictor corrector based algorithm to solve the nonlinear precipitation–dissolution models. For equidistribution approach, we choose an adaptive monitor function and smooth it based on a diffusive mechanism. Numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method by examples through finite difference approach for 1D and finite element approach in 2D. The moving mesh refinement accurately resolves the front location of Darcy scale precipitation–dissolution reactive transport model and reduces the computational cost in comparison to numerical simulations using a fixed mesh.

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“…To cite a few, for significant research contributions toward linear SPDEs mostly with discontinuous convection coefficient, one can recall the articles [2, 22–27] where solution of SPDE possesses strong interior layers and the articles [28–31] where solution of SPDE generates both boundary and strong interior layers. In this regard, we cite the recent research finding in Shiromani et al [32] for 2D elliptic singularly perturbed convection‐diffusion problems with discontinuous convection and source terms, which give rise to strong interior layer phenomena.…”

Section: Introductionmentioning

confidence: 95%

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

Yadav,

Mukherjee

2024

Math Methods in App Sciences

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This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

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A finite difference method for a singularly perturbed 2‐D elliptic convection‐diffusion PDEs on Shishkin‐type meshes with non‐smooth convection and source terms

Cited by 10 publications

References 46 publications

Assessment of Buckling Failure in Oil Storage Tanks: Finite Element Simulation of Combined Internal and External Pressure Scenarios

Assessment of Buckling Failure in Oil Storage Tanks: Finite Element Simulation of Combined Internal and External Pressure Scenarios

Adaptive mesh based efficient approximations for Darcy scale precipitation–dissolution models in porous media

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

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