Back
Prologue
This is a sequel to the present author’s Feature Story, “Mathematics in civil engineering: set theory, algebra and analysis,” published in the December 2024 issue (Volume 52) of Hong Kong Engineer. This Feature Story’s purpose is to elucidate concisely certain mathematical concepts in nonlinear dynamical systems, numerical methods and stochastic methods, together with their applications in geotechnical engineering modelling.
Collaboration between geotechnical engineer and mathematician
Chaos theory
A chaotic dynamical system—given its apparent randomness—has intrinsic complex deterministic patterns, interconnection, constant feedback loops, repetition, self-similarity (namely, that the component part resembles the whole), self-organisation (crystal growth for example), and so on. A chaotic dynamical system is sensitive to initial values and nonlinear in nature. Moreover, it moves from order to disorder, or from stability to instability. As an aside, the defining characteristics of a chaotic system include stochasticity, high sensitivity to initial conditions, aperiodicity, and boundedness.
The butterfly effect, a fundamental principle in chaos theory, describes a phenomenon in which a small alteration in the state of a chaotic dynamical system will cause subsequent states to differ greatly from the states that would have obtained without the alteration. Chaos theory is used, for example, in the following literature:
(i) Gao H D et al (2005). ‘Lateral displacement prediction of deep excavation combining chaos theory with credible regions’, Journal of Beijing University of Technology, 2(31), pp. 136-140.
(ii) Huang Z Q et al (2009). ‘The chaotic characteristics of landslide evolution: A case study of Xintan landslide’, Environmental Earth Sciences , 56(8), pp. 1585-1591. (in Three Georges of China)
Bifurcation theory
A bifurcation occurs in a dynamical system when a small smooth change in certain parameter values (the bifurcation parameters) of the system causes a sudden “qualitative” change in its behaviour. Bifurcation can lead to chaos, but chaos can also arise without bifurcation. For example, in structural mechanics, buckling is a mathematical bifurcation problem.
Bifurcation can also be found in triaxial shear test of soil samples. At the end of the test, the soil material in the cylinder fails and forms sliding regions within itself, which are known as shear bands. The shear band contains the transition from a homogeneous deformation pattern to a heterogeneous one with the appearance of a system of localised zones or bands with excessive shear strain within them. The occurrence of shear band is mathematically treated as a bifurcation problem; that is to say, the constitutive relations may allow the homogeneous deformation pattern to lead to a bifurcation point, at which a heterogeneous deformation pattern with a jump of planar strain rate is admissible.
Bifurcation Theory is used, for example, in the following literature:
(i) Lu X L et al (2011). ‘Strength of soils considering the influence of deformation bifurcation under true triaxial condition’, Yantu Lixue/Rock and Soil Mechanics , 32(1), pp. 21-26.
(ii) Feng P C et al (2022). ‘Influence of bifurcated fracture angle on mechanical behaviour of rock blocks’, Indian Geotechnical Journal , 53, pp. 622-633.
Catastrophe theory
Catastrophe theory is a branch of bifurcation theory. Bifurcation occurs in a parametric dynamical system when a change in a parameter causes an equilibrium to split into two; catastrophe occurs when the stability of an equilibrium breaks down, causing the system to jump into another state. In other words, small changes in certain parameters (or bifurcation points) of the system can cause equilibrium to appear or disappear, or to change from attracting (stable) to repelling (unstable or neutral), and vice versa.
Catastrophe theory is used, for example, in the following literature:
(i) Zhang Y M et al (2008). ‘Estimation of liquefaction based on catastrophe theory’, Chinese Journal of Computational Mechanics , 25(2), pp. 237-240.
(ii) Zhou Z H et al (2020). ‘Stability of rock slope with bedding intermittent joints based on catastrophe theory’, Journal of China Coal Society, 45(S1), pp. 161-172.
Dissipative structure
A dissipative system is defined as an open system that relies on external energy flows to maintain its organisation and will dissipate energy in the process. A tornado can be regarded as a dissipative system.
Dissipative structures are open energy systems with thermodynamic steady states. Living organisms are considered as dissipative structures. These structures are nonlinear.
Dissipative structure is used, for example, in the following literature:
(i) Song D Z et al (2012). ‘Rock burst prevention based on dissipative structure theory’, International Journal of Mining Science and Technology, 22(2), pp. 159-163. (Rock burst is a phenomenon in deep rock tunnels)
(ii) Nicot F et al (2013). ‘Shear banding as a dissipative structure from a thermodynamic viewpoint’, Journal of the Mechanics and Physics of Solids , 179, 105394.
A hybrid modelling approach is conductive to risk reduction in designs concerning slopes of great consequence
Discrete element method
The Discrete element method (DEM) is usually used to simulate the movement of granular materials. It achieves this through calculations that trace the individual particles that constitute the granular material. It is considered as a particle-based method.
A granular flow is essentially a mixture of a granular (particle) material and one fluid, for example, air or water. It is an energetic flow of particles and fluids where grain-to-grain interactions dominate. Sand transport on riverbeds, landslides and debris flow are some examples of granular flows.
The DEM is used, for example, in the following literature:
(i) Albaba A (2016). Discrete element modelling of the impact of granular debris flows on rigid and flexible structures. PhD thesis, Université Grenoble Alpes.
(ii) Li L C et al (2024). ‘DEM analysis of installation and bearing process of open-ended piles considering plugging effects’, Chinese Journal of Geotechnical Engineering, 46(7), pp. 1471-1480.
Particle finite element method
The particle finite element method (PFEM) uses a finite element mesh to discretise the physical domain and to integrate the differential governing equations. In contrast to classical finite element approximations, the nodes of the mesh in the PFEM move according to the equations of motion, which is a Lagrangian approach. Since the nodes can move, large deformation problem can be handled in the PFEM, but not by classical FEM.
PFEM is used, for example, in the following literature:
(i) Zhang X et al (2020). ‘A case study and implication: particle finite element modelling of the 2010 Saint-Jude sensitive clay landslide’, Landslides , 17, pp. 1117-1127. (Saint-Jude is in Canada).
(ii) Carbonell J M et al (2022). ‘Geotechnical particle finite element method for modelling of soil-structure interaction under large deformation conditions’, Journal of Rock Mechanics and Geotechnical Engineering, 14(3), pp. 967-983.
Kriging
Geostatistics is a branch of statistics that focuses on spatial or spatiotemporal data. Kriging is a geostatistical technique that interpolates the value of a geotechnical data within an area (for example, the thickness of colluvium) at an unobserved location from observations of its value at nearby locations. It is based on a Gaussian process governed by prior covariances among values of various locations.
It is a linear model, which uses linear weights that depend not only on the distance between points but also on the direction and orientation of the local points. The algorithm is as follows:
(a) The spatial covariance structure of the sampled points is determined by fitting a variogram. The variogram is defined as a function that describes the degree of spatial dependence between observations as a function of distance. (b) Weights derived from this covariance structure are used to interpolate values for unobserved points across the spatial field.
Kriging is used, for example, in the following literature:
(i) Chala A and Ray R P (2023). ‘Generation and evaluation of CPT data using Kriging interpolation technique’, Periodica Polytechnica Civil Engineering, 67(2), pp. 545-551. (ii) Huang S Y and Liu L L (2024). ‘New Kriging methods for efficient system slope reliability analysis considering soil spatial variability’, Reliability Engineering & System Safety, 245, 109989.
Random field
A random field is a collection of random variables, (x1, x2..), one for each point in the field. Every point in the field (site) xi is a random variable, and all points are mutually correlated to varying degrees.
It is used, for example, in the following literature:
(i) Zhou X P et al (2019). ‘Probabilistic Analysis of Step-Shaped Slopes Using Random Field Models’, International Journal of Geomechanics , 20(1), 04019145.
(ii) Zhang X L et al (2021). ‘Random field model of soil parameters and the application in reliability analysis of laterally loaded pile’, Soil Dynamics and Earthquake Engineering, 147, 106821.
Bayesian inference
To recapitulate, this method uses prior knowledge, in the form of a prior distribution, in order to estimate posterior probabilities. But you may ignore its importance in research. Bayesian inference is used, for example, in the following literature:
(i) Tian H M et al (2021). ‘Efficient and flexible Bayesian updating of embankment settlement on soft soils based on different monitoring datasets’, Acta Geotechnica, 17(1), pp. 1-22.
(ii) Contreras L F (2020). Bayesian methods to treat geotechnical uncertainty in risk‑based design of open pit slopes. PhD Thesis, The University of Queensland.
Non-parametric methods
Non-parametric methods, or distribution-free methods, are statistical methods that do not rely on assumptions about the data being drawn from a given probability distribution. These methods are often applied when less information is known about the data, so that a probability distribution cannot be assumed. A histogram is an example of a non-parametric estimate of a probability distribution. An example of non-parametric test is chi-square test of independence, which determines whether there is an association between categorical variables (that is to say, whether the variables are independent or related). Non-parametric methods are used, for example, in the following literature:
(i) Yun H B and Reddi L N (2011). ‘Nonparametric monitoring for geotechnical structures subject to long-term environmental change’, Advances in Civil Engineering, 2011, pp. 1-17.
(ii) Barani et al (2019). ‘A non-parametric approach to site- and soil-specific probabilistic seismic hazard analysis’ in Silvestri F and Moraci N (ed.) Earthquake Geotechnical Engineering for Protection and Development of Environment and Constructions , London: CRC Press.
Epilogue
Readers should have, by now, become more cognizant of the vast applications of advanced mathematical concepts to multifaceted topics in geotechnical engineering modelling. Nevertheless, it should not be presumed that there are no more mathematical concepts that can be used in geotechnical or civil engineering modelling. In fact, many more helpful mathematical concepts abound, such as machine learning algorithms, stochastic process, and so on. It is the author’s wish that they could be introduced in the future.
About the author
Ir Lau Wai Hin, Lincoln is a Corporate Member of the HKIE (Civil and Geotechnical Disciplines) who holds a Bachelor of Science in Civil Engineering (honours), a Master of Philosophy in Engineering, and a Postgraduate Certificate in Education (Mathematics), all from The University of Hong Kong. He is now a practising Civil and Geotechnical Engineer in a local consultant firm and has over 25 years of engineering experiences.