Computer simulations continue to have an increasing importance in engineering design. Despite advances in computing power, simulation models that capture complex physical phenomena at high fidelities remain intractable for design tasks requiring repeated evaluations of quantities of interest. In these cases, engineers can benefit from developing design insights---a deep understanding of the underlying design space through visualisation and isolating physically significant input parameters.

In this thesis, we explore the use of orthogonal polynomial ridge approximations to construct surrogate models. These models enable rapid evaluations and facilitate a deeper, physically-intuitive understanding of the quantities of interest. For modelling smoothly varying functions, the use of orthogonal polynomials has seen considerable research that establishes its efficacy for surrogate modelling, especially in applications such as uncertainty quantification. Meanwhile, ridge approximations provide low-dimensional representations of functions via techniques from model-based dimension reduction. By expressing high-dimensional functions via a much smaller parameter space, ridge approximations enable the visualisation of functional behaviour and can massively speed up computations.

The interplay between these two classes of methods is explored through novel algorithms presented in this thesis. First, the application of polynomial ridge approximations to the sensitivity analysis of parameterised models is studied. It is shown that polynomial ridge approximations are competitive against other sparse approximation methods for evaluating moment-based sensitivity indices. Moreover, a polynomial-based set of indices---extremum Sobol' indices---is proposed for extremum sensitivity analysis, which is aimed at revealing input parameters responsible for driving the output near extrema. These indices are compared against those based on skewness decomposition, revealing qualitative similarities.

Following this, new methods to form ridge approximations for multi-objective (vector-valued) functions are proposed and studied. The focus of this study is on how ridge approximations of individual objectives can be combined while accounting for relations between the multiple outputs under different situations. First, assuming the underlying objectives represent a scalar field (such as a fluid flow field or finite element mesh), physical properties of the underlying field can facilitate ridge approximations of the field itself and associated quantities of interest. In this work, heuristics grouped under embedded ridge approximations are proposed, which allow emulators of scalar fields to be constructed with efficiency in both computational load and storage space. Using embedded ridge approximations, the storage size of the pressure field around an airfoil can be reduced by over 70% with a small mean squared error of 0.002 times the output variance.

Second, techniques for keeping multiple objectives constant through invariant subspaces are proposed. While the finding of invariant subspaces has been established for single objectives as a consequence of existing ridge approximation methods, the extension of this to multiple objectives is relatively new. Depending on the characteristics of the objectives, two different methods---the intersection approach and the vector-valued approach---are discussed. Both of these methods are then applied as part of the computational backbone for blade envelopes, which is a novel computational framework for designing performance-based tolerance bounds of bladed components in turbomachinery. During manufacturing and in-service use, blades inevitably suffer from geometric deviations away from design intent. Blade envelopes provide a guideline for judging whether geometric variations are likely to lead to performance degradation without explicitly solving for the resultant flow field. They are created by quantifying the statistical distribution of permissible geometries first through sampling parametric multi-objective invariant subspaces, which are then lifted into the parameter-free geometric space via the Mahalanobis distance. The resultant framework is able to accommodate multiple scalar- and vector-valued objectives, and assist the robust design of blades through enabling inverse design and visualisation.