Computational Gradient Elasticity and Gradient Plasticity with Adaptive Splines

Classical continuum mechanics theories are largely insufficient in capturing size effects observed in many engineering materials: metals, composites, rocks etc. This is attributed to the absence of a length scale that accounts for microstructural effects inherent in these materials. Enriching the classical theories with an internal length scale solves this problem. One way of doing this, in a theoretically sound manner, is introducing higher order gradient terms in the constitutive relations. In elasticity, introducing a length scale removes the singularity observed at crack tips using the classical theory. In plasticity, it eliminates the spurious mesh sensitivity observed in softening and localisation problems by defining the width of the localisation zone thereby maintaining a well-posed boundary value problem. However, this comes at the cost of more demanding solution techniques. Higher-order continuity is usually required for solving gradient-enhanced continuum theories, a requirement difficult to meet using traditional finite elements. Hermitian finite elements, mixed methods and meshless methods have been developed to meet this requirement, however these methods have their drawbacks in terms of efficiency, robustness or implementational convenience. Isogeometric analysis, which exploits spline-based shape functions, naturally incorporates higher-order continuity, in addition to capturing the exact geometry and expediting the design-through-analysis process. Despite its potentials, it is yet to be fully explored for gradient-enhanced continua. Hence, this thesis develops an isogeometric analysis framework for gradient elasticity and gradient plasticity. The linearity of the gradient elasticity formulation has enabled an operator-split approach so that instead of solving the fourth-order partial differential equation monolithically, a set of two partial differential equations is solved in a staggered manner. A detailed convergence analysis is carried out for the original system and the split set using NURBS and T-splines. Suboptimal convergence rates in the monolithic approach and the limitations of the staggered approach are substantiated. Another advantage of the spline-based approach adopted in this work is the ease with which different orders of interpolation can be achieved. This is useful for consistency, and relevant in gradient plasticity where the local (explicit formulation) or nonlocal (implicit formulation) effective plastic strain needs to be discretised in addition to the displacements. Using different orders of interpolation, both formulations are explored in the second-order and a fourth-order implicit gradient formulation is proposed. Results, corroborated by dispersion analysis, show that all considered models give good regularisation with mesh-independent results. Compared with finite element approaches that use Hermitian shape functions for the plastic multiplier or mixed finite element approaches, isogeometric analysis has the distinct advantage that no interpolation of derivatives is required. In localisation problems, numerical accuracy requires the finite element size employed in simulations to be smaller than the internal length scale. Fine meshes are also needed close to regions of geometrical singularities or high gradients. Maintaining a fine mesh globally can incur high computational cost especially when considering large structures. To achieve this efficiently, selective refinement of the mesh is therefore required. In this context, splines need to be adapted to make them analysis-suitable. Thus, an adaptive isogeometric analysis framework is also developed for gradient elasticity and gradient plasticity. The proposed scheme does not require the mesh size to be smaller than the length scale, even during analysis, until a localisation band develops upon which adaptive refinement is performed. Refinement is based on a multi-level mesh with truncated hierarchical basis functions interacting through an inter-level subdivision operator. Through Bezier extraction, truncation of the bases is simplified by way of matrix multiplication, and an element-wise standard finite element data structure is maintained. In sum, a robust computational framework for engineering analysis is established, combining the flexibility, exact geometry representation, and expedited design-through analysis of isogeometric analysis, size-effect capabilities and mesh-objective results of gradient-enhanced continua, the standard convenient data structure of finite element analysis and the improved efficiency of adaptive hierarchical refinement

Decarbonising heating and cooling using temperature setback and geothermal energy

The lion’s share of buildings’ energy consumption is used for maintaining a thermally comfortable indoor environment. Strategies of reducing heating and cooling demand can thus be crucial for buildings to achieve net zero. This research aims to investigate the extent to which an occupancy-based temperature setback strategy and geothermal energy supply can decarbonise an office building. The objectives include: 1) exploring the optimal setback temperature for maximum energy savings, both in present time and under the future climate scenarios, and 2) evaluating the extent to which a geothermal borehole can meet the building’s energy demand. The outcome shows that a temperature setback strategy coupled with geothermal energy supply can decarbonise heating and cooling by around half. As for overall building energy demand, temperature setback can make demand reduction by over a tenth while the geothermal energy can meet the demand by a minimum of a fifth.</p

Investigating scalability of deep borehole heat exchangers: numerical modelling of arrays with varied modes of operation

Deep Borehole Heat Exchangers (DBHEs) are a potentially important method of developing geothermal resources through closed-loop systems for carbon neutral, spatial heating. Past research has primarily focused on single-well systems, with few investigating arrays of multiple DBHEs as a method of extracting more thermal energy. In this study, a series of arrays were modelled using OpenGeoSys software, with the aim of understanding the influence of array geometry, inter-borehole spacing and the mode of operation on the thermal performance and system efficiency. OpenGeoSys software is a finite-element model which solves thermal fluxes through the wellbore and surrounding rock using the dual-continuum method. Simulations were undertaken for the lifetime of an array (20 years) with modes of operation testing 1) long-term constant heat load application and 2) intermittent operation with 6 months of extraction followed by a recovery period. Results indicate geometry and mode of operation had a significant impact on inter-borehole spacing and system performance. For long term constant heat load application of 50 kW per DBHE, the minimal spacing required for line and square arrays should be 40 and 30 m. When considering intermittent operation, recovery periods allow replenishment of heat around the borehole, meaning smaller spacing can be utilised

Assessing the technical potential for underground thermal energy storage in the UK

Heating and cooling both make up a large part of the total energy demand in the UK; long-term seasonal thermal energy storage (STES) can address temporal imbalances between varying supply and demand of heat to buildings and processes. Underground thermal energy storage (UTES) can play a role in energy decarbonisation by storing waste heat from space cooling, refrigeration, data processing, industrial processes, harvested summer solar thermal energy or even heat generated by surplus renewable (solar or wind) electricity with fluctuating supply. This paper evaluates a range of UTES technologies in a UK context and addresses geological suitability, storage capacity, low-carbon heat sources, surface heat sources and demand. This review concludes that there is a significant potential for UTES in the UK for both aquifer thermal energy storage (ATES) and borehole thermal energy storage (BTES) systems, coinciding with surface heat sources and demand. Therefore, uptake in UTES technology will help achieve net-zero carbon neutral targets by 2050. There is also scope to utilise UTES technologies within existing subsurface infrastructure. There are 464 oil and gas wells which could be repurposed upon end of life using different UTES technologies. However, the potential for repurposing needs further evaluation; deep single well BTES systems will have a high surface area to volume ratio for storage, reducing the efficiency of such systems and the potential for ATES is limited by issues associated with contaminants. 23,000 abandoned mines underlay ~25 % of the UKs population and could be utilised for minewater thermal energy storage (MTES)

Convergence analysis of Laplacian-based gradient elasticity in an isogeometric framework

A convergence study is presented for a form of gradient elasticity where the enrichment is through the Laplacian of the strain, so that a fourth-order partial differential equation results. Isogeometric finite element analysis is used to accommodate the higher continuity required by the inclusion of strain gradients. A convergence analysis is carried out for the original system of a fourth-order partial differential equation. Both global refinement, using NURBS, and local refinement, using T-splines, have been applied. Theoretical convergence rates are recovered, except for a polynomial order of two, when the convergence rate is suboptimal, a result which also has been found for the (fourth-order) Cahn-Hilliard equation. The convergence analyses have been repeated for the case that an operator split is applied so that a set of two (one-way) coupled partial differential equations results. Differences occur with the results obtained for the original fourth-order equation, which is caused by the boundary conditions, which is the first time this effect has been substantiated

Spatial Analyses of Suitable Solid Waste Dumping Sites in Damaturu, Yobe State Nigeria

The generation and disposal of solid waste is a serious problem in urban areas especially in developing countries. This is because of high generation rates, insufficient budget and machinery for solid waste management, inappropriate techniques and few or non-existent suitable dumping sites. The main objective of this study is to propose suitable areas for solid waste dumping in Damaturu Town, which are environmentally suitable and economically viable. The main data used for this study were Landsat 8 OLI TIRS image with a spatial resolution of 30m; digital elevation model (DEM) with 30m spatial resolution, and ground control point (GCP) collected with a handheld global positioning system (GPS). The maps were prepared by overlay and suitability analysis was carried out using geographic information system (GIS), remote sensing techniques and multi criteria analysis methods. The final suitability map was produced by overlay analyses in ArcMap and levelled as high, moderate, less suitable, and unsuitable regions. The results indicate that 65% of the study area is unsuitable for solid waste dumping; 1.3% less suitable; 21.8% moderately suitable; and 11.9% most suitable. The potential most suitable areas for solid waste dumping sites fall on southern, south eastern and south western parts of the town where there are least environmental and health risks. The GIS and remote sensing techniques are important tools for solid waste site selection. Hence, the capacity to use GIS and remote sensing technology for the effective identification of suitable solid waste dumping site will minimize the environmental risk and human health problems

An isogeometric analysis approach to gradient-dependent plasticity

Gradient-dependent plasticity can be used to achieve mesh-objective results upon loss of well-posedness of the initial/boundary value problem because of the introduction of strain softening, non-associated flow, and geometric nonlinearity. A prominent class of gradient plasticity models considers a dependence of the yield strength on the Laplacian of the hardening parameter, usually an invariant of the plastic strain tensor. This inclusion causes the consistency condition to become a partial differential equation, in addition to the momentum balance. At the internal moving boundary, one has to impose appropriate boundary conditions on the hardening parameter or, equivalently, on the plastic multiplier. This internal boundary condition can be enforced without tracking the elastic-plastic boundary by requiring urn:x-wiley:nme:media:nme5614:nme5614-math-0001-continuity with respect to the plastic multiplier. In this contribution, this continuity has been achieved by using nonuniform rational B-splines as shape functions both for the plastic multiplier and for the displacements. One advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher, making it consistent with the interpolation of the plastic multiplier. This is different from previous approaches, which have been exploited. The regularising effect of gradient plasticity is shown for 1- and 2-dimensional boundary value problems