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Hydrothermal ore‐forming systems usually involve coupled processes among pore‐fluid flow, heat transfer, mass transport and chemical reactions in fluid‐saturated porous media. The finite element method (FEM) has provided a powerful tool to solve hydrothermal ore‐forming problems associated with pore‐fluid convective flow. Depending on different interests of research, a hydrothermal ore‐forming system may be simulated at two different length‐scale models, such as a regional (i.e., large length‐scale) model or a deposit (i.e., small length‐scale) model. However, there is an unsolved issue to guarantee the consistency between the applied boundary conditions to a deposit model and the predicted results, at exactly the same location as the boundary of the deposit model, from using the regional model. This paper proposes a FEM‐based dual length‐scale simulation procedure to address this issue. In the proposed procedure, a regional model of coarse mesh is first simulated by the FEM. Since a deposit model is located within the regional model, the applied boundary conditions to the deposit model of fine mesh can be determined from the simulation results of the regional model, so that the consistency can be guaranteed at the common boundary between the deposit model and the regional model. Main outcomes of this study have demonstrated that: (1) the proposed FEM‐based dual length‐scale simulation procedure is suitable and applicable for solving coupled problems involving pore‐fluid flow, heat transfer, mass transport and chemical reactions in fluid‐saturated porous media; (2) the proposed procedure is predictive because it is unnecessary to know the location of an ore deposit prior to the simulation of a hydrothermal ore‐forming system; (3) the boundary condition of a deposit model of fine mesh is consistent with the simulation results obtained from the regional model of coarse mesh, so that the reliability of the simulation results obtained from the deposit model can be ensured. [ABSTRACT FROM AUTHOR]

Skarn-hosted deposits are commonly recognized as the consequence of magma intrusion within the Earth's upper crust. The Ergu Pb-Zn deposit can be regarded as a typical skarn-hosted deposit in the hydrothermal ore-forming system within the central Lesser Xing'an Range (LXR), Heilongjiang, China. Although extensive studies were conducted to understand the ore-forming system associated with the Ergu Pb-Zn deposit through using the traditional geoscience methods, the ore-forming process involved in this deposit has not been justified in a strictly scientific manner to date. In this paper, the hydrothermal ore-forming process of the Ergu Pb-Zn deposit is computationally simulated through using the dual length-scale approach associated with the finite element method (FEM). The related computational simulation results have demonstrated that: (1) the pore-fluid convection provides continuous ore-forming fluid and material sources for the Ergu Pb-Zn deposit at the quartz-Pb-Zn sulfide stage; (2) the convective flow of the pore fluid is the main dynamic mechanism, which controls the temperature, chemical species and pore-fluid velocity distributions in the Ergu Pb-Zn deposit; (3) the localized structure plays a key role in controlling the localized pore-fluid flow pattern, which can further control the location and formation of the orebody grade in the Ergu Pb-Zn deposit; (4) the dual length-scale approach associated with the FEM is very useful in dealing with the computational simulation of the hydrothermal ore-forming mechanism involved in the Ergu Pb-Zn deposit. [ABSTRACT FROM AUTHOR]

This paper proposes a computational simulation procedure for simulating chemical dissolution-front instability problems, in which radially divergent flow is involved in fluid-saturated porous media. In the proposed computational simulation procedure, a combination of the finite element and finite difference methods is used to simulate a chemical dissolution-front instability problem involving radially divergent flow, while a new algorithm is used to apply a small perturbation to the problem. Particular attention is paid on simulating low-order modes of an unstable circular chemical dissolution-front propagating in a fluid-saturated porous medium, in which dissolvable materials only occupy a small part, so that the final porosity is remarkably smaller than unity when dissolvable materials are completely dissolved in the chemical dissolution system. To verify the proposed computational simulation procedure, analytical solutions for a benchmark chemical dissolution-front instability problem involving radially divergent flow are derived in a purely mathematical manner. The related computational simulation results have demonstrated that: (1) the proposed computational simulation procedure is correct and useful for simulating chemical dissolution-front instability problems, which are associated with both stable and unstable chemical dissolution systems involving radially divergent flow in fluid-saturated porous media; (2) the simulated shapes of the second-order, third-order and fifth-order modes associated with an unstable chemical dissolution-fronts are respectively an ellipse, a star of three angles and a star of five angles in the unstable chemical dissolution system involving radially divergent flow in the fluid-saturated porous medium; (3) although the heterogeneity of a porous medium can affect the propagation speed of a chemical dissolution-front, it does not affect the low-order mode shape in the unstable chemical dissolution system involving radially divergent flow in the fluid-saturated heterogeneous porous medium. [ABSTRACT FROM AUTHOR]

Purpose: The objective of this paper is to establish a solution strategy for obtaining dual solutions, namely trivial (conventional) and nontrivial (unconventional) solutions, of coupled pore-fluid flow and chemical dissolution problems in heterogeneous porous media. Design/methodology/approach: Through applying a perturbation to the pore-fluid velocity, original governing partial differential equations of a coupled pore-fluid flow and chemical dissolution problem in heterogeneous porous media are transformed into perturbed ones, which are then solved by using the semi-analytical finite element method. Through switching off and on the applied perturbation terms in the resulting perturbed governing partial differential equations, both the trivial and nontrivial solutions can be obtained for the original governing partial differential equations of the coupled pore-fluid flow and chemical dissolution problem in fluid-saturated heterogeneous porous media. Findings: When a coupled pore-fluid flow and chemical dissolution system is in a stable state, the trivial and nontrivial solutions of the system are identical. However, if a coupled pore-fluid flow and chemical dissolution system is in an unstable state, then the trivial and nontrivial solutions of the system are totally different. This recognition can be equally used to judge whether a coupled pore-fluid flow and chemical dissolution system involving heterogeneous porous media is in a stable state or in an unstable state. The proposed solution strategy can produce dual solutions for simulating coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media. Originality/value: A solution strategy is proposed to obtain the nontrivial solution, which is often overlooked in the computational simulation of coupled pore-fluid flow and chemical dissolution problems in fluid-saturated heterogeneous porous media. The proposed solution strategy provides a useful way for understanding the underlying dynamic mechanisms of the chemical damage effect associated with the stability of structures that are built on soil foundations. [ABSTRACT FROM AUTHOR]

Purpose: The objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media. Design/methodology/approach: The porosity, horizontal and vertical components of the pore-fluid velocity and solute concentration are selected as four fundamental unknown variables for describing chemical dissolution-front instability problems in fluid-saturated porous media. To avoid the use of numerical integration, analytical solutions for the property matrices of a rectangular element are precisely derived in a purely mathematical manner. This means that the proposed finite element method is a kind of semi-analytical method. The column pivot element solver is used to solve the resulting finite element equations of the chemical dissolution-front instability problem. Findings: The direct use of horizontal and vertical components of the pore-fluid velocity as fundamental unknown variables can improve the accuracy of the related numerical solution. The column pivot element solver is useful for solving the finite element equations of a chemical dissolution-front instability problem. The proposed semi-analytical finite element method can produce highly accurate numerical solutions for simulating chemical dissolution-front instability problems in fluid-saturated porous media. Originality/value: Analytical solutions for the property matrices of a rectangular element are precisely derived for solving chemical dissolution-front instability problems in fluid-saturated porous media. The proposed semi-analytical finite element method provides a useful way for understanding the underlying dynamic mechanisms of the washing land method involved in the contaminated land remediation. [ABSTRACT FROM AUTHOR]

[Display omitted] • To computationally simulate the ore-forming process in the Luziyuan deposit. • To determine the boundary conditions of the deposit model in a scientific way. • To identify the controlling ore-forming mechanism in the Luziyuan deposit. • To use computational simulations for identifying the location of an ore deposit. The giant Luziyuan Pb-Zn deposit is a typical hydrothermal ore-forming deposit in the southern Baoshan block of SE Tibet. Although extensive studies were conducted to understand the ore-forming system associated with the Luziyuan Pb-Zn deposit through using the traditional geoscience methods, the ore-forming process involved in this deposit has not been justified in a strictly scientific manner to date. In this paper, the hydrothermal ore-forming process of the Luziyuan Pb-Zn deposit is computationally simulated through using the dual length-scale approach associated with the finite element method (FEM). In the dual length-scale approach, the FEM is first used to simulate the regional model of a large length-scale in the geological region where the Luziyuan Pb-Zn deposit is located. Afterward, the FEM is then used to simulate the deposit model of the Luziyuan Pb-Zn deposit. Particularly, the boundary conditions of the deposit model are consistently determined from using the computational simulation results of the regional model. The related numerical results demonstrated that: (1) the pore-fluid convection provides a continuous source of ore-forming fluid and ore-forming material for the Luziyuan Pb-Zn deposit; (2) the convective pore-fluid flow is the main dynamic mechanism, which controls the temperature, pore-fluid velocity and chemical species distribution in the Luziyuan Pb-Zn deposit; (3) the localized structure is the main controlling factor for the localized pore-fluid flow pattern, which further controls the formation and location of the orebody in the Luziyuan Pb-Zn deposit; (4) the dual length-scale approach associated with the FEM is very useful for simulating the hydrothermal ore-forming process of the Luziyuan Pb-Zn deposit. [ABSTRACT FROM AUTHOR]

Summary: This paper presents a semianalytical approach for solving first‐order perturbation (FOP) equations, which are used to describe dissolution‐timescale reactive infiltration instability (RII) problems in fluid‐saturated rocks. The proposed approach contains two parts because the chemical dissolution reaction divides the whole problem domain into two subdomains. In the first part, the interface‐condition substitution strategy is used to derive the analytical expressions of purely mathematical solutions for the FOP equations in the upstream subdomain, where the dissolution chemical reaction is ceased and the FOP equations are weakly coupled. In the second part, the finite element method (FEM) is used to derive the analytical expressions of numerical solutions for the FOP equations in the downstream subdomain, where the dissolution chemical reaction needs to be considered and the FOP equations are strongly coupled so that it is impossible to derive purely mathematical solutions for them. Particular attention is paid to the development of the element‐by‐element forward marching strategy, which is associated with the use of the FEM for solving this new kind of scientific problem. The related analytical results demonstrated that (1) both the dynamic characteristic of a reactive infiltration system and the dimensionless wavenumber can have pronounced influences on the distribution of the FOP dimensionless acid concentration within the entire domain of the dissolution‐timescale RII problems in fluid‐saturated rocks and (2) the FOP dimensionless acid concentration distribution exhibits two significantly different patterns in the upstream and downstream subdomains of the dissolution‐timescale RII system. [ABSTRACT FROM AUTHOR]

Abstract Convective seepage flow plays a key role not only in naturally forming mineral deposits and oil reservoirs, but also in the carbon-dioxide sequestration in fluid-saturated rocks. This paper presents a steady-state numerical solver, which is based on the finite element method (FEM) and progressive asymptotic approach procedure (PAAP), to solve steady-state convective seepage flow problems in fluid-saturated heterogeneous rocks. Although this kind of flow problem is commonly solved by using transient-state numerical solvers, the use of the proposed steady-state numerical solver, which is the main characteristic of this study, can have certain advantages. After the proposed steady-state numerical solver is verified by a benchmark testing problem, for which analytical solutions are available for comparison, it is further applied to simulate steady-state convective seepage flow in the Australia North West Shelf basin, which is composed of naturally-formed heterogeneous porous rocks. Through the simulated numerical results, it can be found that: (1) compared with the analytical solutions of the benchmark testing problem, the use of the steady-state numerical solver can produce correct and accurate numerical simulation results for dealing with convective seepage flow problems in fluid-saturated heterogeneous rocks. (2) The main advantage of using the steady-state numerical solver is to avoid the need of considering initial conditions of the problem, which are commonly difficult to be determined because the convective seepage flow within the upper crust is a past event in geology. (3) In the Australia North West Shelf basin, the convective seepage flow and temperature distribution patterns depend strongly on both the permeability contrast of the faults and the basin geometry (including the layer structures and fault locations) in the computational model. Highlights • To present and verify the numerical method for modeling large-scale convective seepage flow in heterogeneous rocks. • To simulate large-scale convective seepage flow in the Australia North West Shelf basin. • Rock permeability heterogeneity can affect large-scale convective seepage flow and temperature patterns. • Basin geometry can also affect large-scale convective seepage flow and temperature patterns. [ABSTRACT FROM AUTHOR]

Numerical modeling has now become an indispensable tool for investigating the fundamental mechanisms of toxic nonaqueous phase liquid (NAPL) removal from contaminated groundwater systems. Because the domain of a contaminated groundwater system may involve irregular shapes in geometry, it is necessary to use general quadrilateral elements, in which two neighbor sides are no longer perpendicular to each other. This can cause numerical errors on the computational simulation results due to mesh discretization effect. After the dimensionless governing equations of NAPL dissolution problems are briefly described, the propagation theory of the mesh discretization error associated with a NAPL dissolution system is first presented for a rectangular domain and then extended to a trapezoidal domain. This leads to the establishment of the finger-amplitude growing theory that is associated with both the corner effect that takes place just at the entrance of the flow in a trapezoidal domain and the mesh discretization effect that occurs in the whole NAPL dissolution system of the trapezoidal domain. This theory can be used to make the approximate error estimation of the corresponding computational simulation results. The related theoretical analysis and numerical results have demonstrated the following: (1) both the corner effect and the mesh discretization effect can be quantitatively viewed as a kind of small perturbation, which can grow in unstable NAPL dissolution systems, so that they can have some considerable effects on the computational results of such systems; (2) the proposed finger-amplitude growing theory associated with the corner effect at the entrance of a trapezoidal domain is useful for correctly explaining why the finger at either the top or bottom boundary grows much faster than that within the interior of the trapezoidal domain; (3) the proposed finger-amplitude growing theory associated with the mesh discretization error in the NAPL dissolution system of a trapezoidal domain can be used for quantitatively assessing the correctness of computational simulations of NAPL dissolution front instability problems in trapezoidal domains, so that we can ensure that the computational simulation results are controlled by the physics of the NAPL dissolution system, rather than by the numerical artifacts. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper primarily deals with the computational aspects of chemical dissolution-front instability problems in two-dimensional fluid-saturated porous media under non-isothermal conditions. After the dimensionless governing partial differential equations of the non-isothermal chemical dissolution-front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non-isothermal chemical dissolution system within two-dimensional fluid-saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non-isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution-front propagation problem in the non-isothermal chemical dissolution system consisting of a fluid-saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore-fluid pressure, and dimensionless chemical-species concentration in a non-isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore-fluid pressure decreases drastically in the non-isothermal chemical dissolution system. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to investigate the effects of different domain shapes in general and trapezoidal domain shape in particular on the morphological evolution of nonaqueous phase liquid (NAPL) dissolution fronts in two-dimensional fluid-saturated porous media. After the governing equations of NAPL dissolution problems are briefly described, the numerical procedure consisting of a combination of the finite element and finite difference methods is used to solve these equations. The related numerical simulation results have demonstrated that: (1) domain shapes have a significant effect on both the propagating speed and the morphological evolution pattern of a NAPL dissolution front in the fluid-saturated porous medium; (2) an increase in the divergent angle of a trapezoidal domain can lead to a decrease in the propagating speed of the NAPL dissolution front; (3) the morphological evolution pattern of the NAPL dissolution front in a rectangular domain is remarkably different from that in a trapezoidal domain of a large divergent angle; (4) for a rectangular domain, the simplified dispersion model, which is commonly used in the theoretical analysis and numerical simulation, is valid for solving NAPL dissolution instability problems in fluid-saturated porous media; and (5) compared with diverging flow (when the trapezoidal domain is inclined outward), converging flow (when the trapezoidal domain is inclined inward) can enhance the growth of NAPL fingers, indicating that pump-and-treat systems by extracting contaminated groundwater might enhance NAPL dissolution fingering and lead to less uniform dissolution fronts. [Copyright &y& Elsevier]

In dealing with chemical-dissolution-front propagation problems in fluid-saturated porous media, the chemical dissolution front represented by the porosity of the medium may have a very steep slope (i.e., a very large porosity gradient) at the dissolution front, depending on the mineral dissolution ratio that is defined as the equilibrium concentration of the dissolved minerals in the pore-fluid to the solid molar density of the dissolvable minerals in the solid matrix. When the mineral dissolution ratio approaches zero, the theoretical value of the porosity gradient tends to infinity at the chemical dissolution front. Even for a very small value of the mineral dissolution ratio, which is very common in geochemical systems, the porosity gradient can be large enough to cause the solution hard to converge when the conventional finite element method is used to solve a chemical dissolution problem in a fluid-saturated porous medium where the pore-fluid is compressible. To improve the convergent speed of solution, a porosity-gradient replacement approach, in which the term involving porosity-gradient computation is replaced by a new term consisting of pore-fluid density-gradient and pressure-gradient computation, is first proposed and then incorporated into the finite element method in this study. Through comparing the numerical results obtained from the proposed approach with the theoretical solutions for a benchmark problem, it has been demonstrated that not only can the solution divergence be avoid, but also the accurate simulation results can be obtained when the proposed porosity-gradient replacement approach is used to solve chemical-dissolution-front propagation problems in fluid-saturated porous media including pore-fluid compressibility. [ABSTRACT FROM AUTHOR]

Abstract: Convective heat transfer associated with the circulation of pore-fluid in porous rocks and fractures within the upper crust of the Earth is substantial when the temperature gradient is sufficiently high. In order to understand the process of Sn-polymetallic mineralization in the Dachang ore district of Guangxi, a finite element method has been used in this study to simulate both pore-fluid flow and heat transfer in this district. On the basis of related geological, tectonic and geophysical constraints, a computational model was established. It enables a computational simulation and sensitivity analysis to be carried out for investigating ore-forming pore-fluid flow and other key factors that may affect hydrothermal ore genesis in the district. The related simulation results have indicated that: (1) permeable fault zones in the Dachang ore district can serve as preferential pathways for pore-fluid flow on a regional-scale; and (2) the pore-fluid flow can affect the salinity distribution. This latter factor is part of the reason why Sn-polymetallic mineralization has taken place in this district. [Copyright &y& Elsevier]

Abstract: This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains. [ABSTRACT FROM AUTHOR]

Abstract: A new finite element scheme is proposed, in this paper, for solving two-dimensional wave propagation problems in multilayered soils resting on a rigid base. The multilayered soils are treated as multiple horizontal layers of lateral infinite extension in geometry. Since these horizontal layers can be truncated by two artificially truncated vertical boundaries, two high-order artificial boundary conditions are applied for propagating the incoming waves from the interior domain into the far field of the system. Both the semi-analytical method and the truncated boundary migration procedure are used to derive the high-order artificial boundary conditions, which are comprised of a physically meaningful dashpot and a generalized energy absorber. The main advantage of using the proposed finite element scheme is that the derived artificial boundary condition can be straightforwardly implemented in the finite element analysis, without violating the band/sparse structure of the conventional finite element equation. The related numerical examples have demonstrated that the proposed finite element scheme is of high accuracy in dealing with wave propagation problems in multiple horizontal layers. [Copyright &y& Elsevier]

Abstract: In this paper, theoretical and numerical methods are used to investigate pore-fluid flow focused temperature distribution patterns around geological faults and cracks of any length-scales in hydrothermal systems. If the far field inflow is uniform and the long axis of an elliptical fault of any length-scale is parallel to the far field inflow direction, a complete set of analytical solutions has been presented for the pore-fluid velocity, stream function and excess pore-fluid pressure around the elliptical fault embedded in fluid-saturated porous media. Because the analytical solutions are explicitly expressed in the conventional Cartesian coordinate system, not only can they be used to gain a theoretical insight into the pore-fluid flow patterns around geological faults and large cracks, but also they can be used as valuable benchmark solutions for validating any numerical methods. After a finite element computational model is validated by comparing the numerical solutions with the present analytical solutions, it is used to investigate pore-fluid flow focused heat transfer around geological faults in hydrothermal systems. Some interesting conclusions in relation to the effects of geological faults on pore-fluid flow focused heat transfer have been made through both the theoretical and the numerical analyses. [Copyright &y& Elsevier]

We use the finite element method to solve coupled problems between pore-fluid flow and heat transfer in fluid-saturated porous rocks. In particular, we investigate the effects of both the hot pluton intrusion and topographically driven horizontal flow on the distributions of the pore-flow velocity and temperature in large-scale hydrothermal systems. Since general mineralization patterns are strongly dependent on distributions of both the pore-fluid velocity and temperature fields, the modern mineralization theory has been used to predict the general mineralization patterns in several realistic hydrothermal systems. The related numerical results have demonstrated that: (1) The existence of a hot intrusion can cause an increase in the maximum value of the pore-fluid velocity in the hydrothermal system. (2) The permeability of an intruded pluton is one of the sensitive parameters to control the pore-fluid flow, heat transfer and ore body formation in hydrothermal systems. (3) The maximum value of the pore-fluid velocity increases when the bottom temperature of the hydrothermal system is increased. (4) The topographically driven flow has significant effects on the pore-fluid flow, temperature distribution and precipitation pattern of minerals in hydrothermal systems. (5) The size of the computational domain may have some effects on the pore-fluid flow and heat transfer, indicating that the size of a hydrothermal system may affect the pore-fluid flow and heat transfer within the system. [Copyright &y& Elsevier]

The finite element method is used to simulate coupled problems, which describe the related physical and chemical processes of ore body formation and mineralization, in geological and geochemical systems. The main purpose of this paper is to illustrate some simulation results for different types of modelling problems in pore-fluid saturated rock masses. The aims of the simulation results presented in this paper are: (1) getting a better understanding of the processes and mechanisms of ore body formation and mineralization in the upper crust of the Earth; (2) demonstrating the usefulness and applicability of the finite element method in dealing with a wide range of coupled problems in geological and geochemical systems; (3) qualitatively establishing a set of showcase problems, against which any numerical method and computer package can be reasonably validated. [Copyright &y& Elsevier]

We use the finite element method to model three-dimensional convective pore-fluid flow in fluid-saturated porous media when they are heated from below. In particular, we employ the particle-tracking technique to mimic the trajectories of particles in three-dimensional fluid flow problems. The related numerical results demonstrated that: (1) The progressive asymptotic approach procedure, which was previously developed for the finite element modelling of two-dimensional convective pore-fluid flow problems, is equally applicable to the finite element modelling of three-dimensional convective pore-fluid flow in fluid-saturated porous media heated from below. (2) The perturbation of gravity at different planes has a significant effect on the pattern of three-dimensional convective pore-fluid flow and therefore, may influence the pattern of orebody formation and mineralization in three-dimensional hydrothermal systems. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the paper an asymptotic formula has been developed to correct the discretization error for the finite element predicted natural frequencies of membrane transverse vibration problems. The general idea behind deriving this asymptotic formula is that, when the finite element size approaches zero, a discretized finite element system approaches a continuous system and the predicted natural frequencies of the system from the finite element analysis therefore approach the exact solutions of the system. Without losing generality, several different finite element mesh patterns have been considered and the same asymptotic formula for correcting the finite element predicted natural frequency has been obtained for all the different mesh patterns because of the uniqueness of the exact solution to the natural frequency of a real structure. The usefulness, effectiveness and efficiency of the present asymptotic formula have been assessed by a simple but critical problem, for which the exact solution is available for comparison. In order to investigate the applicability of the asymptotic formula to practical engineering problems, two challenging membrane vibration problems of irregular shapes, an L-shape and a tapered shape with a circular hole in the centre, have also been analysed. The related numerical results have demonstrated that the asymptotic formula provides a very useful post-processing error corrector for the finite element predicted natural frequencies of membrane transverse vibration problems, even though the problem domains are of irregular shape. The greatest advantage in using the present asymptotic formula is that it yields a solution of higher accuracy, by simply using the formula to correct the rough solution obtained from a much coarser finite element mesh with fewer degrees of freedom, without any further finite element calculation. [ABSTRACT FROM AUTHOR]