A study on numerical methods for modelling the viscoelastic deformation of the earth due to surface loading

Abstract

The finite element (FE) method has been widely used to simulate the glacial isostatic adjustment (GIA) process. Although seismological observations show that mantle material is compressible, many GIA models are developed by assuming it is incompressible to avoid instability caused by the combined effect of self-gravitation and dilatation. In order to improve the accuracy of the prediction of GIA made by the FE model, the effect of compressibility must be included. The objective of this study is to develop a FE model using the commercial FE package - ABAQUS for simulating the viscoelastic deformation of a self-gravitating compressible earth induced by surface loading. In order to validate the proposed FE model, results are compared with those obtained by the conventional normal mode (NM) method.

The first part of this study aims to use the NM method to calculate the load induced responses for benchmarking with the FE model. For incompressible models, the semi-analytical method (SAM) is used to achieve this purpose. Because no analytical solution has been found for multi-layer compressible models, results can only be calculated by the numerical integration method (NIM), which is less reliable due to numerical errors. The size of numerical errors associated with the NIM are quantified by comparing modes and residues of incompressible models calculated by the NIM with those obtained by the SAM. The time domain responses are computed for different background earth models. Results show that if a lithosphere is included in an earth model, the time when gravitational instability become dominant will take much longer than that for an earth model without a lithosphere. A series of analysis are carried out to choose the most suitable compressible background earth model for computing load induced responses as benchmark.

The second part of this thesis is about the development of FE model in ABAQUS. Wu (2004) demonstrated that the viscoelastic deformation of an incompressible earth can be modelled by ABAQUS, via an iterative stress transformation scheme (IST). Attempts were made to extend the IST scheme to incorporate the effect of compressibility, but results do not agree with the benchmark. The reason of failure is found, and limitation of the IST scheme is revealed. A proof is given to show that if the IST scheme is applied with compressible material, the outputs of ABAQUS will also be transformed.

An alternative approach is proposed to achieve the objective of this study. Benchmark tests are carried out on incompressible and compressible earth models. Good agreement is obtained between the results calculated by the proposed approach and those obtained by the conventional normal mode method. By using the new approach, the spatial distribution of the body force terms is plotted. It is found that for compressible earth model without a lithosphere, the internal buoyancy force due to dilatation could be larger than the sum of supportive forces and trigger overturning of the supportive forces.