Several types of singularities which threaten the integrity of numerical solutions occur in poten... more Several types of singularities which threaten the integrity of numerical solutions occur in potential flow problems. The singularity at the tip of a cutoff wall is strong and cannot be mistreated without unsatisfactory results. Special elements can be used to represent the ...
Several types of singularities which threaten the integrity of numerical solutions occur in poten... more Several types of singularities which threaten the integrity of numerical solutions occur in potential flow problems. The singularity at the tip of a cutoff wall is strong and cannot be mistreated without unsatisfactory results. Special elements can be used to represent the analytical behavior ...
The boundary element method has been successfully applied to groundwater flow problems with stoch... more The boundary element method has been successfully applied to groundwater flow problems with stochastic boundary conditions and forcing functions. The solution system is now extended to cases with random hydraulic conductivity using a perturbation technique.
Publisher Summary This chapter discusses two perturbation boundary element codes for steady groun... more Publisher Summary This chapter discusses two perturbation boundary element codes for steady groundwater flow in heterogeneous aquifers. The boundary element method (BEM) has received increased attention as a method capable of solving problems containing material heterogeneity. The search for the efficient codes continues. The new technique expands the potential into a perturbation series and solves the resultant Laplace and Poisson equations, using the existing BEM codes. The method is highly efficient for slow to moderately varying hydraulic conductivities. For problems with hydraulic conductivity varying over several orders of magnitude within the domain, however, this technique failed to provide a converged solution. The chapter presents two BEM codes that seek to circumvent the convergence problems in the original formulation. The first method adheres to the earlier formulation, but now incorporates a zoning concept. This is to permit the use of slow varying functions for the hydraulic conductivity within each zone while, the variability throughout the entire flow field is large. In this approach, the numerical process involves domain integrations.
A family of global interpolation functions (GIF) is used to represent distributed effects, such a... more A family of global interpolation functions (GIF) is used to represent distributed effects, such as material heterogeneity, recharge, and leakage, in a boundary element simulation of non-homogeneous flows through porous media. The technique is applicable to both deterministic and stochastic flow problems. The family of functions includes polynomial, trigonometric, exponential, and wavelet bases. When the distributed influence (e.g. leakage from adjoining formations) is also a part of the problem being solved, the solution process is iterative. In other cases (e.g. accretion due to rainfall-evaporation) the solution is non-iterative. A number of problems are solved using the new approach, and the results are compared with analytical and semi-analytical solutions.
For Laplace's equation and other homogeneous elliptic equations, when the particular and fund... more For Laplace's equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coeffi-cients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Work-shops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there still exists a gap of con-vergence and errors between computation and theory. In this paper, convergence analysis and error estimates are explored for Laplace's equations with the solution u ∈ H k (k > 1 2), to achieve polynomial convergence rates. Such a basic theory is im-portant for TM and MFS and ...
Using a global interpolation function (GIF) approach, boundary element solutions are obtained for... more Using a global interpolation function (GIF) approach, boundary element solutions are obtained for flows in porous media with random hydraulic conductivity. The formulation is based on the indirect approach. The solution field is decomposed into two parts. The first is due to the influence of the constant (or effective) property of the medium. This is represented by a boundary integral term. The second, which is due solely to the random nature of the permeability, is represented by a series of bases functions. Appropriate expressions are obtained for the statistics of the primary flow variables (hydraulic head and flux) in terms of the statistics of the permeability field. Numerical implementations include the use of different families of orthogonal trigonometric, polynomial, and wavelet bases functions. The ensuing code can be used for both the forward solution (i.e., determination of flow field) and the inverse solution (e.g., identification of parameters).
Abstract In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heur... more Abstract In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered. The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.
This paper presents a Boundary Element algorithm for solving axisymmetric diffusion equation in t... more This paper presents a Boundary Element algorithm for solving axisymmetric diffusion equation in the Laplace transform domain. The free-space Green function of the governing equation is not available in closed form. The problem is resolved by using a combination of methods, including series expansion, interpolation from numeric tables, and direct numerical integration. The solution in the time domain is found by an approximate Laplace inversion.
Stochastic Environmental Research and Risk Assessment, 2005
Abstract Four examples are investigated for the optimal and sustainable extraction of groundwater... more Abstract Four examples are investigated for the optimal and sustainable extraction of groundwater from a coastal aquifer under the threat of seawater intrusion. The objectives and constraints of these management scenarios include maximizing the total volume of water pumped, ...
International Journal of Rock Mechanics and Mining Sciences, 2007
ABSTRACT Heat extraction from enhanced geothermal systems (EGS) can greatly affect the behavior o... more ABSTRACT Heat extraction from enhanced geothermal systems (EGS) can greatly affect the behavior of joints and other discontinuities in the reservoir. Fracture permeability can change in response to fluid injection/extraction, rock cooling, variations of stress field, and mineral dissolution/precipitation. The reduction in effective stress caused by pore pressure increase can cause the slippage of discontinuities, thus inducing seismicity. Studies have shown that thermal stresses generated by cold water injection have a similar effect. In order to assess the influence of thermal stresses on fracture opening and slippage, a 3-D coupled heat extraction/thermal stress/elastic displacement discontinuity model is used in this study. The effects of each mechanism on fracture slip is estimated with particular reference to the Coso geothermal field. The results indicate that under typical field conditions, a substantial increase in fracture slip is observed when thermal stresses are taken into account. The temporal evolution of the thermal stresses suggests that the rock mass deformation will not cease upon stoppage of water injection, which can be a cause of delayed seismic activity.
The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Us... more The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Using Biot theory of poroelasticity, the solution of the Mandel problem is extended to include material transverse isotropy, as well as the compressibility of the pore fluid and the solid constituents of the soil–rock skeleton. Le résultat classique de Mandel, associé à l'effet bien comnnu de Mandel—Cryer, est réexamineé. En utilisant la théorie poro-é1astique de Blot, la solution du probléme de Mandel est étendue afin d'introduire l'anisotropie transverse du matériau, ainsi que la compressibilité des fluides présents dans les pores et celle des constituants solides du squelette du sol ou de la roche.
Applying a polymeric coating on steel members has been demonstrated to be an effective countermea... more Applying a polymeric coating on steel members has been demonstrated to be an effective countermeasure for the threat of blast on building and other structures. The development of a blast-resistant coating for steel with good characteristics in a fire would provide buildings with protection against explosions and a fire following the blast, as well as against ordinary building fires. An important characteristic of this coating would be to provide a thermal barrier to heat, such that the temperature of the steel remains for as long as possible below temperatures at which its strength would be weakened. This paper presents a numerical evaluation of melting and burning of nano-enhanced polymeric coatings applied on steel members. Viscosity measurements were performed to obtain needed parameters for the simulations. Polyurea nanocomposite residues showed a minimum in viscosity with temperature, possibly caused by cross-linking and charring. Model results for the polyurea residue having the lowest minimum viscosity value showed that the coating remained attached to the steel sides, although there was some flow that led to the breaking off of a chunk of material from an overhang.
In the numerical solution of three-dimensional boundary value problems, the matrix size can be so... more In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer's capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson's equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson's equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.
Computers & Mathematics with Applications, 2002
Compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context... more Compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill-conditioning problems when using the classical radial basis functions. However, the support scaling factor and slow convergence rate of the CS-RBFs have also raised issues on the effectiveness of the CS-RBFs. In this paper, two multilevel schemes have been proposed to alleviate these problems.
This paper employs particle modeling (PM) for the simulation of dynamic fragmentation in a 2D pol... more This paper employs particle modeling (PM) for the simulation of dynamic fragmentation in a 2D polymeric material (nylon-6,6) subject to the impact of a rigid indenter. A new particle interaction scheme (nearest-second neighboring particle interaction) is raised in an attempt to eliminate a mesh bias in the direction of fracture propagation that a regular lattice model with uniform axial linkage possesses in common. The modeling results compare favorably with the according experimental observations, in terms of the impact load and energy profiles, the specimen deflection value at the peak of the load, and the eventual crack pattern. A mesh dependence problem of PM is discovered while using different mesh resolutions and preliminarily investigated. Furthermore, a number of investigations are conducted accounting for other parameters: (i) sharpness of indenter (flat, small or large curvature), (ii) different drop velocity of indenter, and (iii) microscale perturbation in the material's property. These investigations show that the dynamic response of the impact and the resultant fracture patterns are dependent on the material's constitutive property, the geometric shape of indenter, and the impact velocity.
Several types of singularities which threaten the integrity of numerical solutions occur in poten... more Several types of singularities which threaten the integrity of numerical solutions occur in potential flow problems. The singularity at the tip of a cutoff wall is strong and cannot be mistreated without unsatisfactory results. Special elements can be used to represent the ...
Several types of singularities which threaten the integrity of numerical solutions occur in poten... more Several types of singularities which threaten the integrity of numerical solutions occur in potential flow problems. The singularity at the tip of a cutoff wall is strong and cannot be mistreated without unsatisfactory results. Special elements can be used to represent the analytical behavior ...
The boundary element method has been successfully applied to groundwater flow problems with stoch... more The boundary element method has been successfully applied to groundwater flow problems with stochastic boundary conditions and forcing functions. The solution system is now extended to cases with random hydraulic conductivity using a perturbation technique.
Publisher Summary This chapter discusses two perturbation boundary element codes for steady groun... more Publisher Summary This chapter discusses two perturbation boundary element codes for steady groundwater flow in heterogeneous aquifers. The boundary element method (BEM) has received increased attention as a method capable of solving problems containing material heterogeneity. The search for the efficient codes continues. The new technique expands the potential into a perturbation series and solves the resultant Laplace and Poisson equations, using the existing BEM codes. The method is highly efficient for slow to moderately varying hydraulic conductivities. For problems with hydraulic conductivity varying over several orders of magnitude within the domain, however, this technique failed to provide a converged solution. The chapter presents two BEM codes that seek to circumvent the convergence problems in the original formulation. The first method adheres to the earlier formulation, but now incorporates a zoning concept. This is to permit the use of slow varying functions for the hydraulic conductivity within each zone while, the variability throughout the entire flow field is large. In this approach, the numerical process involves domain integrations.
A family of global interpolation functions (GIF) is used to represent distributed effects, such a... more A family of global interpolation functions (GIF) is used to represent distributed effects, such as material heterogeneity, recharge, and leakage, in a boundary element simulation of non-homogeneous flows through porous media. The technique is applicable to both deterministic and stochastic flow problems. The family of functions includes polynomial, trigonometric, exponential, and wavelet bases. When the distributed influence (e.g. leakage from adjoining formations) is also a part of the problem being solved, the solution process is iterative. In other cases (e.g. accretion due to rainfall-evaporation) the solution is non-iterative. A number of problems are solved using the new approach, and the results are compared with analytical and semi-analytical solutions.
For Laplace's equation and other homogeneous elliptic equations, when the particular and fund... more For Laplace's equation and other homogeneous elliptic equations, when the particular and fundamental solutions can be found, we may choose their linear combination as the admissible functions, and obtain the expansion coeffi-cients by satisfying the boundary conditions only. This is known as the Trefftz method (TM) (or boundary approximation methods). Since the TM is a meshless method, it has drawn great attention of researchers in recent years, and Inter. Work-shops of TM and MFS (i.e., the method of fundamental solutions). A number of efficient algorithms, such the collocation algorithms, Lagrange multiplier methods, etc., have been developed in computation. However, there still exists a gap of con-vergence and errors between computation and theory. In this paper, convergence analysis and error estimates are explored for Laplace's equations with the solution u ∈ H k (k > 1 2), to achieve polynomial convergence rates. Such a basic theory is im-portant for TM and MFS and ...
Using a global interpolation function (GIF) approach, boundary element solutions are obtained for... more Using a global interpolation function (GIF) approach, boundary element solutions are obtained for flows in porous media with random hydraulic conductivity. The formulation is based on the indirect approach. The solution field is decomposed into two parts. The first is due to the influence of the constant (or effective) property of the medium. This is represented by a boundary integral term. The second, which is due solely to the random nature of the permeability, is represented by a series of bases functions. Appropriate expressions are obtained for the statistics of the primary flow variables (hydraulic head and flux) in terms of the statistics of the permeability field. Numerical implementations include the use of different families of orthogonal trigonometric, polynomial, and wavelet bases functions. The ensuing code can be used for both the forward solution (i.e., determination of flow field) and the inverse solution (e.g., identification of parameters).
Abstract In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heur... more Abstract In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered. The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.
This paper presents a Boundary Element algorithm for solving axisymmetric diffusion equation in t... more This paper presents a Boundary Element algorithm for solving axisymmetric diffusion equation in the Laplace transform domain. The free-space Green function of the governing equation is not available in closed form. The problem is resolved by using a combination of methods, including series expansion, interpolation from numeric tables, and direct numerical integration. The solution in the time domain is found by an approximate Laplace inversion.
Stochastic Environmental Research and Risk Assessment, 2005
Abstract Four examples are investigated for the optimal and sustainable extraction of groundwater... more Abstract Four examples are investigated for the optimal and sustainable extraction of groundwater from a coastal aquifer under the threat of seawater intrusion. The objectives and constraints of these management scenarios include maximizing the total volume of water pumped, ...
International Journal of Rock Mechanics and Mining Sciences, 2007
ABSTRACT Heat extraction from enhanced geothermal systems (EGS) can greatly affect the behavior o... more ABSTRACT Heat extraction from enhanced geothermal systems (EGS) can greatly affect the behavior of joints and other discontinuities in the reservoir. Fracture permeability can change in response to fluid injection/extraction, rock cooling, variations of stress field, and mineral dissolution/precipitation. The reduction in effective stress caused by pore pressure increase can cause the slippage of discontinuities, thus inducing seismicity. Studies have shown that thermal stresses generated by cold water injection have a similar effect. In order to assess the influence of thermal stresses on fracture opening and slippage, a 3-D coupled heat extraction/thermal stress/elastic displacement discontinuity model is used in this study. The effects of each mechanism on fracture slip is estimated with particular reference to the Coso geothermal field. The results indicate that under typical field conditions, a substantial increase in fracture slip is observed when thermal stresses are taken into account. The temporal evolution of the thermal stresses suggests that the rock mass deformation will not cease upon stoppage of water injection, which can be a cause of delayed seismic activity.
The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Us... more The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Using Biot theory of poroelasticity, the solution of the Mandel problem is extended to include material transverse isotropy, as well as the compressibility of the pore fluid and the solid constituents of the soil–rock skeleton. Le résultat classique de Mandel, associé à l'effet bien comnnu de Mandel—Cryer, est réexamineé. En utilisant la théorie poro-é1astique de Blot, la solution du probléme de Mandel est étendue afin d'introduire l'anisotropie transverse du matériau, ainsi que la compressibilité des fluides présents dans les pores et celle des constituants solides du squelette du sol ou de la roche.
Applying a polymeric coating on steel members has been demonstrated to be an effective countermea... more Applying a polymeric coating on steel members has been demonstrated to be an effective countermeasure for the threat of blast on building and other structures. The development of a blast-resistant coating for steel with good characteristics in a fire would provide buildings with protection against explosions and a fire following the blast, as well as against ordinary building fires. An important characteristic of this coating would be to provide a thermal barrier to heat, such that the temperature of the steel remains for as long as possible below temperatures at which its strength would be weakened. This paper presents a numerical evaluation of melting and burning of nano-enhanced polymeric coatings applied on steel members. Viscosity measurements were performed to obtain needed parameters for the simulations. Polyurea nanocomposite residues showed a minimum in viscosity with temperature, possibly caused by cross-linking and charring. Model results for the polyurea residue having the lowest minimum viscosity value showed that the coating remained attached to the steel sides, although there was some flow that led to the breaking off of a chunk of material from an overhang.
In the numerical solution of three-dimensional boundary value problems, the matrix size can be so... more In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer's capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson's equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson's equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.
Computers & Mathematics with Applications, 2002
Compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context... more Compactly supported radial basis functions (CS-RBFs) have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill-conditioning problems when using the classical radial basis functions. However, the support scaling factor and slow convergence rate of the CS-RBFs have also raised issues on the effectiveness of the CS-RBFs. In this paper, two multilevel schemes have been proposed to alleviate these problems.
This paper employs particle modeling (PM) for the simulation of dynamic fragmentation in a 2D pol... more This paper employs particle modeling (PM) for the simulation of dynamic fragmentation in a 2D polymeric material (nylon-6,6) subject to the impact of a rigid indenter. A new particle interaction scheme (nearest-second neighboring particle interaction) is raised in an attempt to eliminate a mesh bias in the direction of fracture propagation that a regular lattice model with uniform axial linkage possesses in common. The modeling results compare favorably with the according experimental observations, in terms of the impact load and energy profiles, the specimen deflection value at the peak of the load, and the eventual crack pattern. A mesh dependence problem of PM is discovered while using different mesh resolutions and preliminarily investigated. Furthermore, a number of investigations are conducted accounting for other parameters: (i) sharpness of indenter (flat, small or large curvature), (ii) different drop velocity of indenter, and (iii) microscale perturbation in the material's property. These investigations show that the dynamic response of the impact and the resultant fracture patterns are dependent on the material's constitutive property, the geometric shape of indenter, and the impact velocity.
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Papers by Alexander H.-D. Cheng