Optimised multi-objective design of weir structures Open Access

Costly expansion of infrastructure is needed to meet increasing water, food and energy demands and to adapt to climate change (HMG, 2011; PwC, 2010; UN, 2020). Among these, diversion structures that are used to raise the water level of a river or a canal are ubiquitous in water supply, irrigation and small-scale hydropower applications (Tschantz, 2014). The number of large dams worldwide is currently estimated to be around 58 000 (Mulligan et al., 2020), with Lehner et al. (2011) estimating there may be more than 16 million smaller impoundments. In the UK alone, there are more than 13 000 weirs (EA, 2010; Kitchen, 2022) – a large number compared with the 2800 UK dams (McKie, 2019).

The design of diversion weir structures involves calculating parameters of the structure's components considering surface and subsurface flows, the nature of the foundation soil, structural stability and cost. A range of surface and subsurface flow theories have been forwarded for the safe and economical design of diversion weir structures on permeable foundations – the creep theories of Bligh (1912) and Lane (1935) are two of the oldest. Pavlovsky (1922) developed a general theory of the conformal transformation and Casagrande (1937) proposed a graphical solution of the Laplace equation for steady state flow. Khosla (1944) proposed the method of independent variables based on the Schwarz–Christoffel transformation (Christoffel, 1867). Khosla's method achieves an approximation of the pressure distribution under the structure and the exit gradient of the seeping water at its downstream by splitting the complex foundation profile into several elementary forms.

Recent works have suggested solving the complex non-linear design problem using optimisation. Singh (2011) presented an optimisation-based procedure for barrages that used a genetic algorithm to minimise overall cost while satisfying safety and functionality requirements. Al-Shukur and Fadhil Hassan (2015) conducted a parametric analysis to investigate the effect of variations in design parameters on the dimensions and overall cost of a barrage. Each parameter was considered separately while keeping the others constant. Al-Juboori and Datta (2018) trained meta-models on multiple datasets of simulated seepage scenarios to assess the effect of uncertainty in seepage due to variations in hydraulic conductivity using coupled the simulation–optimisation method. Safety factors and other hydraulic design requirements were imposed as constraints of the optimisation within the simulation model. Ohadi and Jafari-Asl (2021) used multi-objective optimisation in the design of a trapezoidal labyrinth weir; they demonstrated that construction materials (cost) and reliability could be improved conjunctively. Tao et al. (2021) used a genetic algorithm to calibrate an artificial intelligence algorithm that was then used to identify the sensitivity of depth scouring to various parameters of a submerged weir. Ferdowsi et al. (2021) showed that concurrent optimisation of an open channel section and a labyrinth weir geometry could reduce the cost of the structures by up to 11% and 74%, respectively.

The literature shows that cost-effective designs can be achieved with optimisation and that many alternative parameter combinations can lead to acceptable designs. However, all these demonstrations are limited to reducing cost through a single objective optimisation and do not explore the trade-offs between the many relevant design objectives, thus failing to reveal possibly superior designs by exploring the full parameter space. The tendency to inadvertently ignore important decision alternatives and thus the substantial potential to improve performance when using low-dimensional optimisation has been recognised by scholars such as Hogarth (1981), Brill et al. (1982) and Woodruff et al. (2013). Visual interaction with results allows engineers and stakeholders to introduce minimum performance requirements (by filtering or ‘brushing’ the results) (Reed and Kollat, 2013). In the ‘generate first–choose later’ approach (e.g. Geressu and Harou, 2015, 2019; Geressu et al., 2022; Herman et al., 2014) enabled by many objective optimisation and visual analytics, assessments are not restricted by a lone designer's assumptions of acceptable performance trade-offs and safety factors against failure, which may vary depending on site characteristics such as

• how critical the diversion structure is to the water system and if it can be maintained in the case of failure

• the political implications of failure for an important structure

• uncertainty with regards to the hydrology and geology of the site (Singh, 2013)

• the anticipated construction quality

• the availability of funds and so on.

The advantage of optimising performance criteria explicitly rather than treating them as constraints is that a design engineer will be able to see the full set of possibilities. The minimum stability performance criteria for structural stability or cost can then be set post-optimisation to identify the most acceptable design.

Design methods for diversion weirs in the literature treat the performance objectives of stability (i.e. stability against sliding, overturning, scour, piping) as constraints. For example, stability against overturning is checked by ensuring that the sum of moment forces stabilising the structure is at least twice the magnitude of the sum of forces that would destabilise it. If any additional cost is deemed affordable, a higher ratio is preferable to lower the risk of failure further, especially in the case of uncertainties such as on the magnitude and frequency/probability of a high flood discharge. Exploring the combination of parameters that would increase each of the stability factors of safety would be advantageous to identify acceptable balances of the multiple performance goals (increasing stability and reducing cost). To achieve this, the design problem was formulated such that the safety factors – which are traditionally treated as constraints – are treated as performance metrics that are to be maximised.

In this work, a heuristic optimisation (Hadka and Reed, 2012; Kollat and Reed, 2006) approach coupled with a simulation model of vertical and horizontal forces on a diversion weir structure was used to explore the design space and find a solution that meets the objectives. The high computational requirements of finite-difference and finite-element methods, which are preferred for their accuracy in calculating the seepage head loss and its distribution (Al-Juboori and Datta, 2018; Tilaye and Hailu, 2020), make their use with many objective evolutionary algorithms computationally challenging. Meta-models (surrogates of computationally expensive simulation models) and simplified approximate equations can be used to bypass such challenges (e.g. Beh et al., 2017; Jin et al., 2003). The method of independent variables (Khosla, 1944) was used for calculation of seepage and exit gradient because of the significant computational savings coupled with its capacity to approximate uplift pressures, especially where the assumptions of isotropic seepage and deep permeable media are justified (Geressu, 2009).

The proposed approach results in multiple alternative designs from which decision makers can choose based on the acceptable trade-offs between metrics of stability (against piping, sliding, overturning and uplift failure) and construction cost. The approach is demonstrated using a sample design. The simulation–optimisation method was used to explore alternative designs by scaling the parameters of the diversion structure's different components from their minimum recommended values. This ‘generate first–choose later’ approach was then used with visual analytics to show the sensitivity of the performance metrics (e.g. cost and exit gradient) to the design parameters (e.g. depth of cut-off piles and length and thickness of the horizontal apron).

A free and open-source code is provided as a convenient design and decision tool that simultaneously provides cost efficiency and higher factors of safety against multiple modes of failure such as sliding, rotation, uplift and piping.

The paper is set out as follows. The traditional method of independent variables proposed by (Khosla, 1944) is revised in Section 2 using a simplified design problem with hypothetical hydrologic and soil properties. The proposed reformulation of the diversion structure design problem is presented in Section 3. The multi-objective optimisation design results are presented in Section 4. The paper ends with a discussion (Section 5) and conclusions (Section 6).

A sample design problem is now introduced to demonstrate the usage and value of the suggested design approach in relation to the traditional design method of diversion structures built on permeable media. A synthetic design problem with a simplified diversion weir structure is used for ease of communication and to facilitate reproducibility of the results. The following parameters are assumed: head = 7 m, safe exit gradient G_ES = 0.125, discharge intensity = 2.5, Lacey's silt factor f = 0.75, density of concrete material (with which the apron and the cut-off piles will be built) $ρc$ = 2240 kg/m³, density of water $ρw$ = 1000 kg/m³ and gravitational acceleration g = 9.81 m/s².

Design of the longitudinal profile of the structure included setting the depths of the upstream and downstream sheet piles and the apron lengths and thicknesses at various points along the structure of the upstream and downstream aprons. The longitudinal profile of the structure was designed by considering the surface and subsurface flows and the geology of the site.

The upstream and downstream sheet piles were designed to guard against anticipated scouring action of surface water. The scour depth (R) depends on soil properties and discharge intensity:

where q is the discharge intensity and f is Lacey's silt factor (Lacey, 1939).

The downstream sheet pile depth is traditionally set to be more than 1.5R below the high flood level and the upstream pile depth is set at 1.25R below the high flood level. The exit gradient is the hydraulic gradient of the seepage flow under the base of the weir floor. The rate of seepage increases with an increase in the exit gradient. Progressive erosion, commonly called piping failure, can be minimised by reducing the exit gradient. The total length of the apron and the depth of the downstream pile act to keep the exit gradient in a safe limit. Khosla (1944) gives the relation between these parameters as:

where G_ES is the safe exit gradient, $λ$ is a  function of the relation between the total apron length (L_t) and the downstream pile depth (d_d) and ΔH is the maximum head difference anticipated from high flood flow, pond level flow and static water cases.

The value of $λ$ for a permissible safe exit gradient (G_ES) and downstream pile depth (d_d) is therefore:

A parameter $α$ is defined as:

and then the total apron length (L_t) is calculated as:

Once the pile depths and the total length of the aprons are set based on the surface flow (i.e. scour depth) and subsurface flow (stability against piping failure), the structure is checked for structural stability (i.e. stability against sliding, overturning, uplifting and crushing due to excessive pressure). It should be noted that the minimum length of the apron downstream of the weir is determined by the length of the hydraulic jump. For this simplified example, it is assumed that the minimum downstream apron length is 10 m.

An uplift force exists on the structure because of the subsurface flow of water underneath it. The uplift pressure head decreases from upstream to downstream. Design for stability against uplift is achieved by balancing the thickness of aprons at various points along the longitudinal section with the uplift pressure due to the subsurface flow. The forces and moments acting on the corresponding structure are then calculated and the structure is checked for stability against overturning and sliding. The apron thickness at a certain point (t) is:

where $Δh$ is the difference between the uplift pressure head and the surface water depth and G is the relative density of the apron construction material. The unbalanced head (h) is the residual head at each point under the apron. The residual head is calculated downstream of the upstream pile and upstream of the downstream pile using Khosla's method of independent variables (Khosla, 1944) (please see online supplementary material (TraditionalDesign.xlsx) at https://github.com/RoTiGe/pyWeir/ for detailed calculations). The residual head at intermediate points is assumed to vary linearly with distance from upstream to downstream.

Calculations of the forces and moments acting on the structure are shown in Table 1.

A factor of safety of 1.5–2.0 is usually applied for safety against overturning (Baban, 1995). In order to avoid lifting up of the structure's heel and tension occurrence at the base, the resultant force must pass through the middle third of the structure's base:

where e is the eccentricity (distance of resultant force from middle of the structure), $Lt$ is the total apron length and X is distance of the application of the resultant force from the toe of the structure.

However, if this condition is not satisfied, the tension and compression at the hill of the weir should be checked to conform with $ρmin$ > 0 and $ρmax$ < 70 t/m²; ρ_min is the tension pressure at the hill of the weir and ρ_max is the compression pressure at the hill of the weir:

The structure may slide in the flow direction if there is not enough grip between the base and the foundation. To prevent this happening, the vertical forces need to be checked to be adequate and compared with the horizontal forces to supply static friction that would keep the structure intact in its place. The US Bureau of Reclamation, as quoted by Baban (1995), suggests ∑V/ ∑H  > 0.35 for concrete structures on common soils, where ∑V is the sum of external vertical forces and ∑H is the sum of external horizontal forces.

The parameters of the components of diversion weir structures have a complex non-linear relationship and multiple parameter combinations lead to designs with varying levels of safety against different failure modes and varying costs.

Approaches that link simulation models with heuristic global search methods such as evolutionary algorithms (Coello et al., 2007; Deb et al., 2002) are well suited for handling the non-linearity present in the design of diversion structures. In this approach, the design goal is to find the combination of parameters (shown in Figure 1) that will achieve the highest stability factors against multiple forms of failure at the least cost. A computer program that calculates the stability performance metrics and cost for a set of input parameters (Table 2) was linked to a multi-objective evolutionary algorithm (MOEA) to reveal a set of Pareto-optimal designs that best balance multiple performance objectives. The proposed multi-objective optimisation approach was validated by comparing the design results with those based on the traditional method, in which the thickness of the horizontal apron is optimised with a parametric grid search. The multi-objective design problem is formulated as:

where f_x is the objective metric for the exit gradient (stability against piping) and f_u, f_o, f_s, f_e and f_c are, respectively, the performance metrics for stability against uplift, stability against overturning, stability against sliding, eccentricity and cost.

The decision variables are the depths of the upstream and downstream cut-off piles and the length and thickness of the horizontal aprons upstream and downstream of the weir position. The range of values each parameter can take has to be provided to the search algorithm.

Results of optimisation using MOEA are stochastic, with no guaranteed achievement of the true Pareto front through a single optimisation run. Pareto-sorting of solutions from different random seeded runs (with different initial points) could better approximate the extent of the true Pareto front (Zatarain Salazar et al., 2017). A lack of improvement in the hypervolume score with more evolutions (number of generations) was taken as the stopping criterion. Different numbers of generations (multiples of 1000) were experimented with, each run with 30 randomly seeded initial parameter sets. The hypervolume score comparison showed no further improvement beyond 10 000 generations and thus the optimisation runs were conducted with 30 random seeds and 10 000 generations (see Appendix for details). The results from each run were then sorted together to provide the best overall reference set (Kollat et al., 2008).

The results of the proposed approach as applied to the stylised weir design problem are described in this section. The range of values each parameter can take has to be provided to the MOEA. The values from the traditional design method discussed in Section 1 were used as indicators of the parameter ranges (see Table 3). The trade-offs between the various performance metrics are first discussed taking two, three and then six of them at time, assuming the unit construction costs for the pile and apron material are equal. The sensitivity of the performance metrics to the relative cost of pile and apron and how they affect the optimal parameters of the structure is then discussed.

Figure 2 shows the performance of the optimised designs and the trade-offs between cost and exit gradient, both of which are to be minimised. The design labelled C shows the possible simultaneous gain compared with the traditional design (labelled A) in both reducing cost and increased safety against failure by piping (represented by the exit gradient) when optimisation is applied. A performance comparison of designs B and C shows how, with a relatively small increase in cost, a large improvement in the exit gradient may be achieved. However, a further decrease in the exit gradient, say from C to D, incurs a substantial cost increase (by more than four-fold). All of the designs performed better than the traditionally designed structure (A) in at least one of the two performance objectives. However, some of the Pareto-optimal designs (e.g. B and C) performed equally well or better in all objectives, meaning they are better choices than design A.

The best designs when only the cost and the exit gradient minimisation objectives are considered may not meet other performance criteria such as stability against uplift, overturning and sliding. The blue circles in Figure 3(a) show designs that are dominated if only cost and exit gradient are the decision criteria but which are relevant when stability against uplift (represented by the size of the markers) is also considered.

Figure 3 shows that the designs that were only Pareto-optimal for cost and exit gradient performed poorly in terms of stability against uplift (shown by the size of the marker in Figure 3(a)). However, a design with a higher performance in stability against uplift (e.g. large circle around B) can be found with a performance close to designs that were Pareto-optimal when considering only cost and exit gradient. Although higher cost is associated with more construction materials, designs with higher cost do not necessarily perform better in terms of stability against failure by uplift because a larger portion of the material goes into the cut-off pile depths (to reduce the exit gradient) in some of the designs (e.g. design E).

Figure 3(b) shows the same information as Figure 3(a) in a different way. The values at the top of the parallel axes represent the highest achievable performance if that particular objective were to be prioritised. The non-vertical lines represent efficient (Pareto-optimal) designs. Lines that cross between two adjacent axes signal a trade-off between those measures. The steepness of the angles indicate the strength of the trade-off between the two performance indicators. The pink markers in Figure 3(a) and the lines in Figure 3(b) compare designs made with using the traditional method with the designs optimised based on the suggested formulation. The result shows (e.g. blue lines in Figure 3(b)) that the multi-objective optimisation produced designs that were superior in all three performance objectives.

Figure 4 shows that the safety factors against uplift and overturning had a relatively small trade-off between themselves but the objectives of minimising cost and the exit gradient showed strong trade-offs. The green lines in Figure 4 show designs that met the minimum performance criteria of stability against uplift (>1), stability against overturning (>2), stability against sliding (>0.35) and an eccentricity of <1/6, which are commonly regarded as acceptable minimum stability performance criteria. The distribution of the green lines on the six right-most axes (representing the range of structural parameters) shows that various combinations of parameter values can lead to acceptable results that both meet the minimum performance criteria and are non-dominated when considering the various performance objectives.

Figure 5 shows how variations in the relative costs of pile and apron construction affect the performance metrics and their relationship with the optimal parameters. For clarity, the figure only shows designs that met the minimum stability criteria.

Figure 5 shows only designs that met the minimum stability criteria. The colour of the lines distinguish the designs based on their cost; the bold solid black lines represent designs with the least 1% cost compared with all the others. Comparing the results from the different pile–apron cost scenarios shows that increasing the upstream pile depth (beyond that which is necessarily to guard against scouring) should not be a priority (i.e. if reducing cost is the primary objective) if the cost of pile construction is high compared with the cost of the apron per unit volume (Figure 5(b)). Increasing the upstream pile depth should be considered only in the situation where the relative pile cost is lower than the cost apron. In all the scenarios, the least-cost design was not necessarily the one that achieved an undesirable threshold exit gradient. The least-cost designs scored just above the minimum requirements for stability against uplift force and overturning in the scenarios where the relative pile cost was double than that of apron. However, designs that achieved substantially better performance in stability against uplift force and overturning were possible when the pile cost was less than the cost of the apron (black lines in Figures 5(a) and 5(c)). In all the relative cost scenarios, increasing the downstream pile length and the total apron length accompanied by keeping the downstream apron length at a minimum resulted in low-cost designs (black and blue lines).

The strongest trade-off was between the exit gradient and cost. Although the stability against overturning showed correlation (the lines are parallel) with stability against uplift and against sliding at the lower half of the performance potential range, the higher half of performance was associated with decreased performance (higher trade-offs with) in stability against uplift and sliding (indicated by the steep slope in the crossing lines).

A comparison of the results shown in Figures 2 and 3 demonstrates how, by using traditional empirical methods or single-objective optimisation, infrastructure design engineers could inadvertently ignore important decision alternatives and the substantial potential to improve performance. In the traditional approach, where often only one design is sought, the design process stops when the minimum criterion for each stability criterion, which are treated as constraints, is met. The results show that, for the same or similar amounts of construction material (and hence cost), a structure could have large differences in the stability metrics. Even when all these metrics are above the minimum recommended, it would be prudent to choose a design with a more balanced distribution of stability metrics against the failure modes. In the newly suggested multi-objective optimisation approach, each stability criterion is considered as a performance objective, which is either maximised (i.e. stability against uplift, sliding and overturning) or minimised (exit gradient). This allows for exploration of alternative designs that best balance the stability criteria and the overall cost. Figure 2 shows how the optimisation search achieved better performing designs (shown with blue circles) compared to the traditional approach (labelled A). There often are uncertainties with measurements such as the safe exit gradient (a function of high flood volume and head, which in turn are uncertain because of potential climate and environmental changes) and the granular composition of the permeable soil on which the head work is built. The suggested multi-objective optimisation and visual analysis approach allows decision makers to decide the level of ‘safety factor’ they want to adopt in view of the additional cost that will involve and the decision maker's ability or willingness to trade these.

Figure 5 shows how optimal design varies based on the relative costs of piles and apron construction costs. Generalisation of the optimal relationship between pile depths, apron lengths and their thickness for different construction sites could thus be misleading. In addition, the costs of materials and construction costs of piles and aprons can vary over time, making an optimum design at one point in time (while in the design stage) sub-optimal during the construction stage.

The manual design of diversion structures, especially with the method of independent variables proposed by (Khosla, 1944) is complex (see the online supplementary material for the Excel-based calculation for the case study example) and requires extensive experience and time. Khosla's method of independent variables is still used extensively by design engineers, at least in some developing countries such as Ethiopia, for whom the tools and computer facilities are too expensive. A larger number of low-head diversion structures are designed and implemented at more local levels by various engineers and government offices. Although applying optimised designs can help to use scarce resources efficiently, avoiding subjectivity of designs and potential errors is yet to be applied because of the limited capacity and lack of exposure. It is hoped that the computationally efficient, free and open software published with this paper will help familiarise the partitioning community with modern approaches of design automation, multi-objective optimisation and trade-off analysis.

This study shows extensive optimisation and sensitivity analysis is necessary to identify designs that best meet performance criteria at reasonable cost based on site-specific scenarios. Even where high-performance computing facilities are lacking, multi-objective optimisation is possible with the use of computationally efficient approaches of simulation–optimisation using approximation approaches such as Khosla's method of independent variables. The suggested automated design allows for efficient experimentation of alternative designs with differing construction materials and site-specific costs. The design of low-head diversion structures is, by its nature, multi-objective (e.g. minimising cost, maximising safety against uplift, overturning, sliding etc.), making it ideal to introduce to the wider practising engineering community for use in trade-off analysis in decision making.

The availability of computational tools and facilities (particularly in developed countries) allows for a more convenient approach to subsurface flow assessment under diversion structures built on permeable media (i.e. through numerical methods such as finite-difference and finite-element methods). Use of the finite-difference method for subsurface flow under low-head diversion structures has been discussed in previous studies (e.g. Geressu, 2009; Tilaye and Hailu, 2020). Khosla's method (Khosla, 1944) was used in this work as it is a computationally efficient approximation of the pressure distribution. The simulation–optimisation approach used in this study requires computing the subsurface flow for thousands of different subsurface geometries (300 000 for this study). This makes the required computational facilities out of reach for the majority of practitioners in developing countries. The purpose of the multi-objective optimisation is to highlight the optimal trade-offs and the sensitivity of performance to changes in design parameters. For practical use of the method, the stability against uplift and exit gradient can be rechecked after the multi-objective optimisation. The suggested multi-objective formulation would work well with either finite-difference or finite-element methods where access to high-performance computing is available.

The design of low-head diversion structures on natural rivers involves analysis of surface flow conditions at pond level and also at high floods (Geressu, 2009) and for different parts of the cross-section (i.e. weir section and under sluice section for sediment flushing). This study optimised a simplified diversion structure problem (where the high flood discharge was used to calculate the scour depth). The force loadings considered in this paper are for pond level conditions only, for brevity and as a proof of concept.

A multi-objective optimisation design approach for diversion weir structures has been proposed and demonstrated. The problem formulation transforms the traditional constraints in the design of diversion structures, such as safety factors against uplift, overturning and sliding, into performance objectives.

The results showed that designs with a higher performance in stability can be found with no or minimal (within 1% of the overall cost) increase to the cost if the stability performance metrics are optimised explicitly and simultaneously in a multi-objective optimisation. However, designs where parameters are set based on empirical generalised observations or where the design is optimised with only a subset of the parameters searched can perform poorly, even though they may meet the minimum criteria. Multi-objective optimisation considering all performance criteria, site-specific conditions (e.g. relative cost of different construction materials) and parameter variations explicitly helps to identify superior designs, which is not possible with the traditional engineering design approach.

The design of diversion structures is complex and requires extensive experience along with a number of tests to craft the right design for a specific project. These requirements are difficult in many developing countries because of limited human, capital, testing and design resources. This often leads to designs with higher safety factors being preferred. However, because of time, resources and methodological limitations to explore a sufficient number of alternatives, these decisions are often subjective and could constitute misplaced compensatory adjustments and unnecessarily high cost. This study has demonstrated how approaches that explicitly explore the trade-offs between the many relevant design objectives can reveal superior designs and also allow for a multi-party consultation to identify acceptable performance trade-off based on different perspectives. Wide application of the freely available (https://github.com/RoTiGe/pyWeir) and easy-to-use design optimisation tool developed in this study could help standardise designs and lower costs by avoiding large and often unwarranted safety factors.

A heuristic optimisation approach (Hadka and Reed, 2012; Kollat and Reed, 2006) that couples a search algorithm with a simulation model of forces on a diversion structure built on a permeable foundation was employed. The ε-NSGAII algorithm (Deb et al., 2000) generates an initial random population of decision variables by exploiting uniform random sampling within user-specified ranges. These variables are then passed as input variables to a water resources simulator that evaluates the performance of the system. The performance information is passed back to the ε-NSGAII algorithm, which evaluates the fitness of the decision variables to produce the next generation of decision variables. Over successive generations of the optimisation run, high-quality solutions are passed into the ε-dominance archive and injected into the population at the beginning of the next run to automatically adjust the search population size (Kollat and Reed, 2006). The ε-dominance archive sorts solutions based on user-specified levels of significant precision (i.e. the minimum change in performance level between solutions for a user to identify them as significantly different).

In this study, a lack of improvement in the hypervolume score with more evolutions (number of generations) was taken as the stopping criterion. Different numbers of generations (multiples of 1000) were experimented with, each run with 30 randomly seeded initial parameter sets. The hypervolume score comparison showed no further improvement beyond 10 000 generations and thus the optimisation runs were conducted with 30 random seeds and 10 000 generations (Figure 6).

The ε-NSGAII algorithm was chosen for its search effectiveness and availability (Kollat and Reed, 2007). An initial population size of 25 was used with a 0.01 ε value for all the performance metrics in the Platypus framework (Hadka, 2018) on a personal laptop computer.

d_d

downstream pile depth

e

eccentricity (distance of resultant force from middle of the structure)

F

objective function for optimisation

F_cv

vertical force due to apron weight

F_wh

horizontal water force

F_uv

uplift force due to seepage pressure

F_wv

vertical water force

f

Lacey's silt factor

f_c

performance metric for cost of the structure

f_e

performance metric for eccentricity

f_o

performance metric for stability against overturning

f_s

performance metric for stability against sliding

f_u

performance metric for stability against uplift

f_x

exit gradient (stability against piping) objective metric

g

gravitational acceleration

G

specific gravity of soil (relative density of construction material for apron)

G_E

exit gradient

G_ES

safe exit gradient

H

maximum head difference anticipated from high flood flow and pond level flow cases

h

unbalanced head between uplift pressure head and surface water depth

h_w

height of water upstream of weir

L_d

downstream apron length

L_d min

minimum downstream apron length

L_t

total apron length

L_u

upstream apron length

M_cv

moment resulting from the weight of the structure

M_uv

uplift moment (measure of the turning effect of a force)

M_wh

moment resulting from horizontal water force

M_wv

moment resulting from vertical force

p₁

uplift pressure at upstream

p₂

uplift pressure at downstream

q

discharge intensity

R

scour depth

S_dal

scale factor for downstream apron length

S_tal

scale factor for total apron length

t

thickness of apron at a point

t₁

thickness of apron at upstream

t₂

thickness of apron at downstream

t_a

thickness of apron at a point along longitudinal section

V_f

vertical force acting perpendicular to bed

X

distance of the application of the resultant force from the toe of the structure

ΔH

difference between upstream and downstream water levels

Δh

difference between uplift pressure head and surface water depth

λ

function of the relation between total apron length, downstream pile depth and safe exit gradient

ρ_c

density of concrete material

ρ_max

compression pressure at hill of weir

ρ_min

tension pressure at hill of weir

ρ_w

density of water

∑F_H

sum of horizontal forces

∑F_v

sum of vertical forces

∑M_d

sum of destabilising moments

∑M_s

sum of moments leading to stability

∑V

sum of vertical forces

∑V_g

sum of downward forces

∑V_u

sum of uplift forces