Structural engineering from an inverse problems perspective

Abstract

The field of structural engineering is vast, spanning areas from the design of new infrastructure to the assessment of existing infrastructure. From the onset, traditional entry-level university courses teach students to analyse structural responses given data including external forces, geometry, member sizes, restraint, etc.-characterizing a forward problem (structural causalities $\to$ structural response). Shortly thereafter, junior engineers are introduced to structural design where they aim to, for example, select an appropriate structural form for members based on design criteria, which is the inverse of what they previously learned. Similar inverse realizations also hold true in structural health monitoring and a number of structural engineering sub-fields (response $\to$ structural causalities). In this light, we aim to demonstrate that many structural engineering sub-fields may be fundamentally or partially viewed as inverse problems and thus benefit via the rich and established methodologies from the inverse problems community. To this end, we conclude that the future of inverse problems in structural engineering is inexorably linked to engineering education and machine learning developments.

Keywords: inverse problems; machine learning; non-destructive testing and evaluation; smart materials and structures; structural design; structural engineering.

Figure 1.

Schematic illustration depicting the forward and inverse problem relationship for a stretched elastic plate with randomized stiffness properties. The forward finite-element model inputs (causalities) are shown as the non-homogeneous stiffness properties while the model output is the displacement field. By contrast, the inverse model aims to estimate the stiffness properties given the displacement field. (Online version in colour.)

Figure 2.

Relationship between structural analysis and design, in which design is the inverse problem of evaluating a suitable structural solution given a set of constraints. Some constraints (such as stability constraints) render themselves more easily to quantitative treatments. (Online version in colour.)

Figure 3.

Directly measured (UoS) and inferred (UCT) specific impulse distributions from studies of blast loading and plate deformation following detonation of spherical explosives, expressed at 100 g (UoS) scale; after [64].

Figure 4.

Directly measured (UoS) and inferred (UCT) specific impulse distributions from studies of blast loading and plate deformation following detonation of cylindrical explosives, expressed at 78 g (UoS) scale; after [64].

Figure 5.

Reconstructions (right column) report probabilistic predictions of local flexural and shear cracking in concrete elements. The colour bars represent the probability of cracks at nodal locations: (a) simulated shear cracking pattern, (b) probabilistic prediction of the shear crack pattern using a convolutional neural network, (c) photo of a flexural crack in an area-sensing skin and (d) probabilistic prediction of the flexural crack using a feedforward neural network. (Online version in colour.)

Figure 6.

An example of vision-based bolt-loosening detection where (a) and (b) are images of a bolted connection taken at different inspection periods. Two loosened steel bolts are shown in the blow-up figures with counterclockwise rotations in their bolt heads. Using a series of image-processing techniques, the differential features caused by the bolt loosening can be identified in (c). Detailed discussion can be found in [215]. (Online version in colour.)

Figure 7.

Examples of EIT and piezoresistive inversion applied to self-sensing nanocomposites. (a) A soft carbon nanofibre/polyurethane (CNF/PU) is deformed by rigid, non-conductive indentors [274]. EIT is then used to image the deformation-induced conductivity changes, and piezoresistive inversion is used to recover the displacement field (multiplied by a factor of 5 for ease of visibility). (b) A hard CNF/epoxy is loaded in tension with a stress raiser at its centre [275]. EIT is again used to image the conductivity change. Lastly, piezoresistive inversion is used to recover the underlying displacement field. With knowledge of the material’s elastic properties, strains and stresses can be spatially mapped. The first principal stress of the guage section is shown here along with comparison with a traditional FEM solution for validation. (Online version in colour.)