Optimal notch shapes for loaded plates

Stress uniformity within the gauge zone of a cruciform specimen significantly affects materials’ in-plane biaxial mechanical properties in material testing. The stress uniformity depends on the load transmission of the cruciform specimen from the fixtures to the gauge zone. Previous studies failed to alter the nature of the load transmission of the geometric features using parametric optimisations. To improve stress uniformity in the gauge zone, we optimised the cross-arms to design a centre-reduced cruciform specimen with topology and shape optimisations. The simulations show that the optimised specimen obtains significantly less stress variation and range in the gauge zone than the optimised specimen under different observed areas, directions, and load ratios of von Mises, S11, S22, and S12. In the quantified gauge zone, a more uniform stress distribution could be generated by optimizing specimen geometry, whose value should be estimated indirectly each time through simulations. We found that topology and shape optimisations could markedly improve stress uniformity in the gauge zone, and stress concentration at the cross-arms intersection. We first optimised the cruciform specimen structure by combining topology and shape optimisations, which provided a cost-effective way to improve stress uniformity in the gauge zone and reduce stress concentration at the cross-arms intersection, helping obtain reliable data to perform large strains in the in-plane biaxial tensile test.

The problem considered here is to determine the shape of a symmetric two-dimensional plate so that the drag of this plate in infinite cavity flow is a minimum. With the flow assumed steady and irrotational, and the effects due to gravity ignored, the drag of the plate is minimized under the constraints that the frontal width and wetted arc-length of the plate are fixed. The extremization process yields, by analogy with the classical Euler differential equation, a pair of coupled nonlinear singular integral equations. Although analytical and numerical attempts to solve these equations prove to be unsuccessful, it is shown that the optimal plate shapes must have blunt noses. This problem is next formulated by a method using finite Fourier series expansions, and optimal shapes are obtained for various ratios of plate arc-length to plate width.

This work relies on constructal design to perform the geometric optimization of the V-shaped pathways of highly conductive materials (inserts) that remove a constant heat generation rate from a body and deliver it to isothermal heat sinks. It is shown numerically that the global thermal resistance of the V-shaped pathway can be minimized by geometric optimization subject to total volume and V-shaped pathways material constraints. Constructal design and genetic algorithm (GA) optimization showed the emergence of an optimal architecture that minimizes the global thermal resistance: an optimal external shape for the assembly of pathways and optimal geometry features for the V-shaped pathway. Parametric study was performed to show the behavior of the minimized global thermal resistance as function of the volume fraction of the V-shaped pathways. First achieved results for ϕ = 0.3 indicated that when freedom is given to the geometry, the thermal performance is improved. Afterward, the employment of GA with constructal design allowed the achievement of the optimal shapes of V-shaped pathways for different volume fractions (0.2 ≤ ϕ ≤ 0.4). It was not realized the occurrence of one universal optimal shape for the several values of ϕ investigated, i.e., the optimal design was dependent on the degrees of freedom and the parameter ϕ and it is reached according to constructal principle of optimal distribution of imperfections.

Mathematical shape optimization of hip prosthesis design

The femoral component of the artificial hip joint implanted in a patient's femur is subjected to a complex set of forces exerted due to normal life activities. It is thought that high values of stress in the cement in a cemented prosthesis can lead to fractures of the cement mantle and loosening of the stem. The incidence of such problems may be diminished by reduction of the fatigue notch factor in the cement, such that stress concentrations are avoided and the crack initiation time maximised. This study describes a method for numerical shape optimisation to determine an optimal shape for the femoral stem of a hip prosthesis in order to minimise the fatigue notch factor in the cement layer at the interfaces with the bone and stem whilst at the same time maintaining or increasing the stress levels in the proximal medial femoral bone to help prevent stress shielding. The method when used to optimise the shape of a stainless steel Charnley stem was found to be extremely efficient and effective. The resulting optimal shape was heavily waisted in the proximal region below the neck but distally was quite similar to the original design. The fatigue notch factors in the cement were reduced by 16% and 19% for medial and lateral cement/stem interfaces, respectively, and by 8% and 2% for the corresponding cement/bone interfaces. The fatigue notch factor in the proximal medial bone was increased by 57% which indicates that the general stress level in this region is markedly increased. Thus the optimised design should increase the fatigue life of the cement and at the same time reduce stress shielding in the proximal bone. Both of these effects may help prevent loosening of the femoral component and hence reduce the need for early revision operations.

A methodology for structural optimisation with damage tolerance constraints

The transition towards a more affordable, reliable, and sustainable energy provision paradigm is one of the main 21st century challenges that humanity must overcome to protect the planet from the harmful effect caused by climate change. The concentration of CO2 in the atmosphere has been dramatically increasing since the pre-industrial era. If the increase of green-house gasses emissions continues unabated, this will bring dramatic consequences for planet Earth, compromising eventually the existence of many species, including the human race. To avoid a climate change catastrophe, the share of primary energy coming from renewable energy resources must increase from around 15% in 2015 to 65% in 2050. This energy transition can not rely solely on few successful technologies (i.e., solar photovoltaic, and wind energy), but it must count on a larger variety of technical solutions that are suitable for a wider range of renewable sources and diversity of circumstances. For instance, renewable thermal energy sources for power generation (i.e., geothermal reservoir, biomass fuel, and concentrated solar radiation), can provide a large portion of the world electricity demand in the future. However, the exploitation of a good portion of these sources strongly depends on the market success of technologies such as the Organic Rankine Cycle (ORC) power system. One of the key aspects to make ORC systems economically competitive, especially at the smaller sizes (⇡ 1 − 50 kW), is the realization of highly efficient turbomachinery components. The fluid-dynamic design of ORC turbomachinery significantly differs from the design of traditional machines (i.e., steam and gas turbines), and this is mainly due to the different thermo-physical properties and gas dynamic behavior of the organic working fluids. This means that design methods devised for standard steam and gas turbomachinery can not be used for turbomachinery operating in the Non-ideal compressible fluid dynamics (NICFD) region. Furthermore, no experimental campaigns have ever been carried out to create a body of empirical knowledge to support the design highly efficient ORC turbomachinery. As a consequence, the entire design process of ORC turbomachinery relies only on the use of advanced CFD software. The current trend is to couple CFD tools with numerical optimization techniques in order to automatically obtain optimal flow passage geometries. In particular, adjoint-based methods have clearly demonstrated to be the only optimization technique capable of tackling the multi-stage turbomachinery design problem, in which thousands of design variable must be concurrently optimized. Therefore, the research documented in this PhD dissertation aimed at extending the adjoint method in order to perform the fully-turbulent fluid-dynamic shape optimization of 3D multi-stage ORC turbomachinery. This document contains an extensive introduction, three main chapters, each documenting a building block towards the accomplishment of the main goal of this PhD project, and a final concluding chapter that summarizes all the research outcomes of this work and proposes future steps for research in this field. The first part of the thesis describes the extension of the RANS equations, the convective numerical schemes, and the viscous numerical schemes to the use of complex thermo-physical laws, so to simulate turbulent flows of components working in the NICFD thermodynamic region. The second part documents the derivation of the adjoint solver in order to resolve shape-optimization design problems for 2D single row of ORC turbomachinery. Finally, the last part reports the extension of the adjoint method to 3D multi-stage turbomachinery design.

High Cycle Fatigue (HCF) is a very important topic in mechanical engineering problems. Almost any machine part contains some changes in cross section or boreholes connected with stress concentration. It is the task of shape optimization to improve the strength of machine parts by changing the design at free boundaries. In the past, some work was done with the so-called classical stress concentration factors just to minimize the mentioned stress concentration, also for fatigue. This paper represents a more sophisticated view by including modern Continuum Damage Mechanics (CDM) to model material behavior in HCF. The central question is to design machinery parts so that maximum lifetime is obtained. The whole context for this is developed in this paper, including non-linear finite element algorithms. The paper concludes with numerical and experimental tests for the shape optimization of a notched tension bar in HCF.

Operating satellites in the very-low-Earth-orbit (VLEO) region has a number of advantages. However, a primary challenge to the sustained operation of VLEO satellites is the significant atmospheric drag resulting from the higher ambient density, which leads to rapid orbital decay and eventual reentry. Therefore, reducing aerodynamic drag is essential for extending the operational lifetime of VLEO satellites. At the same time, it is important to consider the internal volume to ensure that the satellite maintains its functional capabilities. This work presents a shape optimization framework for VLEO satellites aimed at minimizing the aerodynamic drag without compromising the internal volume. The optimization is performed using a genetic algorithm, with drag force calculated via the panel method and validated by the direct simulation Monte Carlo simulations. Under a fixed-volume constraint, the optimal satellite geometry achieves a maximum drag reduction of 60.3% compared to the reference satellite. Furthermore, this study employs multi-objective optimization to simultaneously minimize aerodynamic drag and maximize internal volume. The results demonstrate that the notable improvements in both drag and volume can be simultaneously achieved through shape optimization. Compared to the reference satellite, the optimal satellite geometry offers a drag reduction of 53.9% and a volume increase of 7.6%.

An optimisation method, based on a genetic algorithm, is applied to derive an optimal tank geometry for enhancement of the roll stability limit of a tank–vehicle combination, specifically under partial fill conditions. The genetic algorithm is applied to minimise the primary overturning moment imposed on the vehicle due to lateral acceleration and cargo load shift expressed in terms of lateral and vertical movement of the liquid bulk within the partly filled tank. The periphery of an arbitrarily shaped tank is generated using B-spline curves to achieve the necessary flexibility to arrive at an optimal shape, where each control point has local influence on the curve. The genetic algorithm considers a combination of the control points as an individual, while each population is formed with a certain number of individuals. The permutations of the individuals are evaluated to derive the best solution that would minimise the objective function based upon the primary overturning moment due to the liquid bulk. The minimisation process is subjected to a number of limit constraints on the resulting total cross-sectional area, overall height and overall width. A static roll plane model of a commercial tank vehicle comprising optimal tank geometry and quasi-static fluid slosh is utilised to study the benefits of the optimal cross-section under various fill conditions in terms of rollover threshold, lateral acceleration and sprung mass roll angle. The potential performance benefits of the proposed optimal tank are discussed in relation to those of the currently used tank cross-sections.

Aerodynamic shape optimization of large-span coal sheds for wind-induced effect mitigation using surrogate models

The optimization of toroidal core shapes for use in self-saturating magnetic amplifiers requires a knowledge of the influence of id-od ratio on apparent core magnetic properties. A theoretical result was derived assuming a parallelogram-shaped basic hysteresis loop and an elaborate experiment was performed. The possibility of doubling the gain of a push-pull amplifier stage by changing the core height and radial build-up to increase the id-od ratio without changing the iron area was shown.

The p method of analysis, when used with shape optimization, has certain advantages over the h method. The method of h adaptivity results in fine meshes with small elements in areas of stress concentration. The presence of these small elements prevents the shape from changing significantly during optimization due to mesh distortion. The p-adaptive method uses relatively large sized elements, and adaptivity is achieved by increasing the polynomial order of the shape functions rather than by introducing small elements as in the h method. Thus, the p-adaptive method allows for greater shape changes without sacrificing the accuracy of the analysis. A methodology is developed in this paper to integrate p-adaptive analysis with shape optimization for general velocity fields. The key contributions lie in an easily implementable shape update strategy and an adaptive scheme to locally increase the polynomial order based on error indicators. Numerical results show that significant shape change is achievable prior to serious mesh distortion. In addition, need for an h-p adaptive strategy rather than a pure p-adaptive strategy is also highlighted.

The notch tensile strength of a boron bearing steel at 1305 MPa has been investigated by means of slow strain rate tests after electrochemical hydrogen charging. Results show that the notch tensile strength of the steel decreased with increasing diffusible hydrogen content and the decrease in the notch tensile strength was more pronounced for specimens at a higher stress concentration factor. Finite element analysis results show that the dependence of the notch tensile strength on stress concentration factor cannot be accounted for by the local peak stress and local peak hydrogen concentration; however, the equivalent plastic strain at the notch root at an applied stress equal to the notch tensile strength was independent of stress concentration factor. The equivalent plastic strain at the notch root can be used as a fracture criterion for hydrogen embrittlement of the steel.

Shape optimization is commonly used in various fields of metal forming. Merklein et al. [6] introduced a novel class of forming process, sheet bulk metal forming (SBMF), which is being continuously improved and applied within the collaborative research group TR73. For SBMF, non-invasive form finding is used to optimize the shape of semi-finished products, which is adapted to the forming process a priori, as it is general practice for tailored blanks. The node-based, non-invasive algorithm minimizes the differences between the spatial configuration and the prescribed discretized target configuration by updating the material configuration iteratively up to the optimum semi-finished product geometry.