Cheng-Yu Ku

On the accuracy of the collocation Trefftz method for solving two- and three-dimensional heat equations

ABSTRACT In this article, the accuracy of the collocation Trefftz method (CTM) for solving two- and three-dimensional heat equations is investigated. The numerical solutions are approximated by superpositioning T-complete functions formulated using cylindrical harmonics. To avoid the ill-conditioning of the CTM, the characteristic lengths and the multiple-scale Trefftz method are adopted. The results reveal that for two-dimensional problems, the CTM can provide highly accurate numerical solutions, with the accuracy increasing with the order of the terms. For three-dimensional problems, highly accurate numerical solutions can be obtained using a certain order of terms, where the order is determined by performing an accuracy assessment.

Solving the Inverse Cauchy Problem of the Laplace Equation Using the Method of Fundamental Solutions and the Exponentially Convergent Scalar Homotopy Algorithm (Ecsha)

In this paper, the inverse Cauchy problem of the Laplace equation is considered. Using the method of fundamental solutions, a system of linear algebraic equations can be obtained by satisfying the Cauchy boundary conditions on the overprescribe boundary points. The resulting linear algebraic equation is ill-posed and is treated by the exponentially convergent scalar homotopy algorithm (ECSHA). Four examples are adopted to show the validity of the proposed numerical scheme and it is concluded that the current approach can successfully resolve the ill-posedness of the inverse Cauchy problem even when the noise exists.

Transient modeling of regional rainfall-triggered shallow landslides

This study investigates the transient modeling of regional rainfall-triggered shallow landslides in unsaturated soil using the Richards equation. To model shallow landslides within a distributed regional-scale framework, infinite slope stability analysis coupled with the hydrological model with consideration of the fluctuation of time-dependent pore water pressure and the soil–water characteristic curve proposed by van Genuchten was developed. The validity of the proposed model is established through several test problems by comparing the numerical results with the analytical solutions. A new procedure to set up wide-range shallow landslide analysis and to integrate regional distribution variations for input data such as geology, groundwater level, hydrogeological characteristics, and rainfall intensity and duration was presented. The results obtained demonstrate that the computed distribution of the safety factor is consistent with the distribution of actual landslides. In addition, the fluctuation of pore water pressure in unsaturated soil dominates the stability of landslides during typhoons accompanied by heavy rainfall. The findings observed in this study are a fundamental contribution to environmental effects for landslides in areas with higher occurrence and vulnerability to extreme precipitation.

Numerical Solutions for Groundwater Flow in Unsaturated Layered Soil with Extreme Physical Property Contrasts

Abstract In this paper, the numerical solutions for groundwater flow in unsaturated layered soil using the Richards equation are presented. A linearisation process for the nonlinear Richards equation to deal with groundwater flow in unsaturated layered soil is derived. To solve one-dimensional flow in the unsaturated zone of layered soil profiles, flux conservation and the continuity of pressure potential at the interface between two consecutive layers are considered in the numerical model. In addition, a novel method, named the dynamical Jacobian-inverse free method, incorporated with a two-side equilibration algorithm for solving ill-conditioned systems with extreme contrasts in hydraulic conductivity is proposed. The validity of the model is established in numerous test problems by comparing the numerical results with the analytical solutions. The results show that the proposed method can improve convergence and numerical stability for solving groundwater flow in unsaturated layered soil with extreme contrasts in hydraulic conductivity.

Rock Mass Hydraulic Conductivity Estimated by Two Empirical Models

Undertaking engineering tasks such as tunnel construction, dam construction, mine development, the abstraction of petroleum, and slope stabilization require the estimation of hydraulic conductivity for fractured rock mass. The understanding of hydraulic properties of fractured rock mass, which involves the fluid flow behaviour in fractured consolidated media, is a critical step in support of these tasks. To obtain hydraulic properties of fractured rock mass, double packer systems can be adopted (NRC 1996). They can be used to determine the hydraulic conductivity in a portion of borehole using two inflatable packers. Although this type of test can directly measure the hydraulic parameter, costs of the testing are fairly high. Several studies (Snow, 1970; Louis, 1974; Carlsson & Olsson, 1977; Burgess, 1977; Black, 1987; Wei et al., 1995;) have proposed the estimation of rock mass hydraulic conductivity using different empirical equations, which were based on the concept that rock mass permeability decreases with depth, as shown in Table 1. These empirical equations provide a great feature for characterizing rock mass hydraulic properties quickly and easily. However, the applicability of these equations is very limited because depth is not the only significant variable on the prediction of rock mass permeability. Hydraulic properties of rock mass may vary with geostatic stress, lithology and fracture properties, including fracture aperture and frequency, fracture length, fracture orientation and angle, fracture interconnectivity, filling materials, and fracture plane features (Lee & Farmer, 1993; Sahimi, 1995; Foyo et al., 2005; Hamm et al., 2007). Thus, a more applicable empirical equation for estimating hydraulic conductivity of rock mass possibly must include the aforementioned factors. This chapter proposes two empirical models to estimate hydraulic conductivity of fractured rock mass. The first empirical model was based on the rock mass classification concept. The study developed a new rock mass classification scheme for estimating hydraulic conductivity of fractured rocks. The new rock mass classification system called as “HCsystem” based on the following four parameters: rock quality designation (RQD), depth index (DI), gouge content designation (GCD), and lithology permeability index (LPI). HCvalues can be calculated from borehole image data and rock core data. The second empirical model was simply based on results of borehole televiewer logging, flowmeter logging and packer hydraulic tests. Three borehole prospecting techniques for fractured rock mass hydrogeologic investigation were performed to explore various hydrogeologic characteristics, such as fracture width, fracture angle, flow velocity and hydraulic

Solving Nonlinear Problems with Singular Initial Conditions Using A Perturbed Scalar Homotopy Method

Abstract In this paper, a novel method, named the perturbed scalar homotopy method, is proposed to solve nonlinear systems with a singular Jacobian matrix. The concept of the proposed perturbed scalar homotopy method roots from the conventional homotopy method but it takes the advantages of converting a vector function to a scalar function by using the square norm of the vector function to conduct a scalar-based homotopy method. Then, a small parameter, which is similar to the perturbation theory, is introduced to the singular systems of nonlinear equations such that the modified singular systems of nonlinear equations become nonsingular and the asymptotic solutions may be found. As a result, the proposed novel method does not need to calculate the inverse of the Jacobian matrix and thus has great numerical stability. In addition, the formulation of the proposed method reveals that this new method is exponentially convergent with the use of the exponential time function. Results obtained show that the proposed novel method can be used to solve singular systems of nonlinear equations with high accuracy as well as the convergence and it may be a better alternative for solving a system of non-linear algebraic equations.

A Novel Boundary-Type Meshless Method for Modeling Geofluid Flow in Heterogeneous Geological Media

A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in the hydraulic conductivity.

Numerical modeling of unsaturated layered soil for rainfall-induced shallow landslides

AbstractIn this paper, a pioneer study on numerical modeling of rainfall-induced shallow landslides in unsaturated layered soil using the variably saturated flow equation is presented. To model the shallow landslides, the infinite slope stability analysis coupled with the hydrological model with the consideration of the fluctuation of time-dependent pore water pressure and Gardner equation for soil water characteristic curve was developed. A linearization process for the nonlinear Richards equation to deal with groundwater flow in unsaturated layered soil is derived using the Gardner model. To solve one-dimensional flow in the unsaturated zone of layered soil profiles, flux conservation and the continuity of pressure potential at the interface between two consecutive layers are considered in the numerical discretization of the finite difference method. The validity of the proposed model is established in three numerical problems by comparing the results with the analytical and other numerical solutions. A...

Integrating Multiple Subsurface Exploration Technologies in Slope Hydrogeologic Investigation: A Case Study in Taiwan

This study aims at presenting an integration of different downhole prospecting techniques for hydrogeologic investigation in an active landslide area. A series of subsurface exploration technologies were conducted, including borehole image scanning, electric logging, groundwater velocity measurements, and double packer testing. Both acoustic and optical borehole loggings as well as electric logging were applied to identify lithology, water bearing capacity and fracturing of the formation around the boring. Subsequently, borehole flow logs were used to indicate the distribution of permeability and hydraulic connectivity of fractures along the borehole. Based on the above prospecting results, test sections of hydraulic tests can be arranged. Finally, hydraulic packer tests were carried out to further characterize the hydrogeologic system of the site and quantitatively determine the hydraulic properties of major hydrogeologic units. Integrating multiple downhole prospecting techniques on slope hydrogeology investigation not only provides hydraulic properties for a study area, but also brings information to establish a hydrogeologic conceptual model and process the model simulation.

Modeling of Transient Flow in Unsaturated Geomaterials for Rainfall-Induced Landslides Using a Novel Spacetime Collocation Method

The modeling of transient flow in unsaturated soils for rainfall-induced landslides using a novel spacetime collocation method is presented. A numerical solution obtained in the spacetime coordinate system is approximated by superpositioning Trefftz basis functions satisfying the linearized Richards equation for collocation points on the spacetime domain boundary. The Gardner exponential model is adopted to derive the linearized Richards equation to describe the soil-water characteristic curve in unsaturated soils. To deal with the rainfall-induced landslides, the infinite slope stability analysis coupled with the proposed meshless method with the consideration of the fluctuation of time-dependent matric suction is developed. The proposed method is validated for several test problems. Application examples of transient modeling of flow for rainfall-induced landslides in homogenous unsaturated soils are also conducted. Numerical results demonstrate that the proposed method is highly accurate to deal with transient flow in unsaturated soils for rainfall-induced landslides. In addition, it is found that the numerical method using the Richards equation with the Gardner model may provide a promising solution for different soil textures.