Local and Dimension Adaptive Sparse Grid Interpolation and Quadrature

In this paper we present a locally and dimension-adaptive sparse grid method for interpolation and integration of high-dimensional functions with discontinuities. The proposed algorithm combines the strengths of the generalised sparse grid algorithm and hierarchical surplus-guided local adaptivity. A high-degree basis is used to obtain a high-order method which, given sufficient smoothness, performs significantly better than the piecewise-linear basis. The underlying generalised sparse grid algorithm greedily selects the dimensions and variable interactions that contribute most to the variability of a function. The hierarchical surplus of points within the sparse grid is used as an error criterion for local refinement with the aim of concentrating computational effort within rapidly varying or discontinuous regions. This approach limits the number of points that are invested in `unimportant' dimensions and regions within the high-dimensional domain. We show the utility of the proposed method for non-smooth functions with hundreds of variables

An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses

A novel approach for non-intrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalised polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multi-element method does not require initial knowledge of the behaviour of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency and generality of the method.Comment: 20 pages, 18 figure

An adaptive minimum spanning tree multielement method for uncertainty quantification of smooth and discontinuous responses

A novel approach for nonintrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalized polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain d

Tolerance analysis and synthesis of assemblies subject to loading with process integration and design optimization tools

Manufacturing variation results in uncertainty in the functionality and performance of mechanical assemblies. Management of this uncertainty is of paramount importance for manufacturing efficiency. Methods focused on the management of uncertainty and variation in the design of mechanical assemblies, such as tolerance analysis and synthesis, have been subject to extensive research and development to date. However, due to the challenges involved, limitations in the capability of these methods remain. These limitations are associated with the following problems: The identification of Key Product Characteristics (KPCs) in mechanical assemblies (which are required for measuring functional performance) without imposing significant modelling demands.  Accommodation of the high computational cost of traditional statistical tolerance analysis in early design where analysis budgets are limited. Efficient identification of feasible regions and optimum performance within the large design spaces associated with early design stages.  The ability to comprehensively accommodate tolerance analysis problems in which assembly functionality is dependent on the effects of loading (such as compliance or multi‐body dynamics). Current Computer Aided Tolerancing (CAT) is limited by: the ability to accommodate only specific loading effects; reliance on custom simulation codes with limited practical implementation in accessible software tools; and, the need for additional expertise in formulating specific assembly tolerance models and interpreting results. Accommodation of the often impractically high computational cost of tolerance synthesis involving demanding assembly models (particularly assemblies under loading). The high computational cost is associated with traditional statistical tolerancing Uncertainty Quantification (UQ) methods reliant on low‐efficiency Monte Carlo (MC) sampling. This research is focused on addressing these limitations, by developing novel methods for enhancing the engineering design of mechanical assemblies involving uncertainty or variation in design parameters. This is achieved by utilising the emerging design analysis and refinement capabilities of Process Integration and Design Optimization (PIDO) tools. ii The main contributions of this research are in three main themes:  Design analysis and refinement accommodating uncertainty in early design;  Tolerancing of assemblies subject to loading; and, efficient Uncertainty Quantification (UQ) in tolerance analysis and synthesis. The research outcomes present a number of contributions within each research theme, as outlined below. Design analysis and refinement accommodating uncertainty in early design: A PIDO tool based visualization method to aid designers in identifying assembly KPCs in early design stages. The developed method integrates CAD software functionally with the process integration, UQ, data logging and statistical analysis capabilities of PIDO tools, to simulate manufacturing variation in an assembly and visualise assembly clearances, contacts or interferences. The visualization capability subsequently assists the designer in specifying critical assembly dimensions as KPCs.  Computationally efficient method for manufacturing sensitivity analysis of assemblies with linear‐compliant elements. Reduction in computational cost are achieved by utilising linear‐compliant assembly stiffness measures, reuse of CAD models created in early design stages, and PIDO tool based tolerance analysis. The associated increase in computational efficiency, allows an estimate of sensitivity to manufacturing variation to be made earlier in the design process with low effort.  Refinement of concept design embodiments through PIDO based DOE analysis and optimization. PIDO tools are utilised to allow CAE tool integration, and efficient reuse of models created in early design stages, to rapidly identify feasible and optimal regions in the design space. A case study focused on the conceptual design of automotive seat kinematics is presented, in which an optimal design is identified and subsequently selected for commercialisation in the Tesla Motors Model S full‐sized electric sedan. These contributions can be directly applied to improve the design of mechanical assemblies involving uncertainty or variation in design parameters in the early stages of design. The use of native CAD/E models developed as part of an established design modelling procedure imposes low additional modelling effort. Tolerancing of assemblies subject to loading:  A novel tolerance analysis platform is developed which integrates CAD/E and statistical analysis tools using PIDO tool capabilities to facilitate tolerance analysis of assemblies subject to loading. The proposed platform extends the capabilities of traditional CAT tools and methods by enabling tolerance analysis of assemblies which are dependent on iii the effects of loads. The ability to accommodate the effects of loading in tolerance analysis allows for an increased level of capability in estimating the effects of variation on functionality.  The interdisciplinary integration capabilities of the PIDO based platform allow for CAD/E models created as part of the standard design process to be used for tolerance analysis. The need for additional modelling tools and expertise is subsequently reduced.  Application of the developed platform resulted in effective solutions to practical, industry based tolerance analysis problems, including: an automotive actuator mechanism assembly consisting of rigid and compliant components subject to external forces; and a rotary switch and spring loaded radial detent assembly in which functionality is defined by external forces and internal multi‐body dynamics. In both case studies the tolerance analysis platform was applied to specify nominal dimensions and required tolerances to achieve the desired assembly yield. The computational platform offers an accessible tolerance analysis approach for accommodating assemblies subject to loading with low implementation demands. Efficient Uncertainty Quantification (UQ) in tolerance analysis and synthesis:  A novel approach is developed for addressing the high computational cost of Monte Carlo (MC) sampling in statistical tolerance analysis and synthesis, with Polynomial Chaos Expansion (PCE) uncertainty quantification. Compared to MC sampling, PCE offers significantly higher efficiency. The feasibility of PCE based UQ in tolerance synthesis is established through: theoretical analysis of the PCE method identifying working principles, implementation requirements, advantages and limitations; identification of a preferred method for determining PCE expansion coefficients in tolerance analysis; and, formulation of an approach for the validation of PCE statistical moment estimates.  PCE based UQ is subsequently implemented in a PIDO based tolerance synthesis platform for assemblies subject to loading. The resultant PIDO based tolerance synthesis platform integrates: highly efficient sparse grid based PCE UQ, parametric CAD/E models accommodating the effects of loading, cost‐tolerance modelling, yield quantification with Process Capability Indices (PCI), optimization of tolerance cost and yield with multiobjective Genetic Algorithm (GA).  To demonstrate the capabilities of the developed platform, two industry based case studies are used for validation, including: an automotive seat rail assembly consisting of compliant components subject to loading; and an automotive switch in assembly in which functionality is defined by external forces and multi‐body dynamics. In both case studies optimal tolerances were identified which satisfied desired yield and tolerance cost objectives. The addition of PCE to the tolerance synthesis platform resulted in large computational cost reductions without compromising accuracy compared to traditional MC methods. With traditional MC sampling UQ the required computational expense is impractically high. The resulting tolerance synthesis platform can be applied to tolerance analysis and synthesis with significantly reduced computation time while maintaining accurac