This paper uses the finite element method to explain the specific nature of numerical instability such as network dependence in the topology optimisation of engineering structures from the perspective of partial differential equations. Gaussian function filtering method reduces the global impact of local extremum on topology optimisation. Finally, the method is introduced into the topology optimisation of concrete gravity dams in hydraulic engineering, and the topology optimisation program is developed in conjunction with ANSYS software language to achieve the topology optimisation of building structures in hydraulic engineering from a technical perspective.

#### Keywords

- ANSYS software
- partial differential equations
- hydraulic engineering
- gravity dam
- building structure
- topology optimisation

#### MSC 2010

- 12H99

Structural topology optimisation is the development of structural shape optimisation and an aspect of layout optimisation. As the shape optimisation gradually matured, the new concept of structural topology optimisation began to develop. Now topology optimisation has become a new hot research topic in the field of international structural optimisation. In the shape optimisation process, the initial structure and the final structure have the same topology. For example, after the shape optimisation of a plate-like structure with three openings, the only change is the boundary shape of the openings. The number of openings has not increased or decreased but is still three. However, there may be a situation where the change in the number of openings may be more effective in reducing the weight of the plate than the change in the shape of the openings under the same design constraints. This is the original research purpose of topology optimisation. Although both finite element and boundary element can be used to automatically divide the mesh, there is no research on the data processing of a model with a topology change, so it is very difficult to automatically generate openings in the design area. To overcome this limitation, consider using a ‘fixed’ finite element model in which the less stressed elements are artificially designated with very soft materials to approximate the openings.

Structural topology optimisation discusses the interconnection methods of structural members, including the topological forms such as the presence or absence of voids and the number and location of holes in the structure, so that the structure can transfer external loads to the support under the constraints of equilibrium, stress and displacement. Seat, and at the same time make certain morphological indicators of the structure optimal. The main difficulty of topology optimisation is that the structural topology that meets certain functional requirements has many or even infinite forms, and this topology form of the target structure is difficult to quantitatively describe or parameterise, and because the area to be designed is unknown in advance, more added difficulty in solving problems. Strictly speaking, logical variables (presence or absence) or integer variables should be used to indicate whether there are rods connected between nodes, but this will complicate the optimisation model; so people are used to starting from the section optimisation model for topology. Optimisation, in simple terms, uses the theory of finite elements to convert topology optimisation into a form of size optimisation for calculation [1].

Based on the above research background, this paper starts from the mathematical theory of topology and in-depth studies the theory of structural topology optimisation and applies it to the practice of hydraulic engineering. The main work is as follows: First, the thesis explains the numerical instability phenomena such as grid dependence in structural topology optimisation from the perspective of the inverse problem of partial differential equations in the proposed structure according to the specific situation of the structural design of hydraulic buildings. In essence, combining the Gaussian function filtering method in digital signal processing and the multigrid method in fluid mechanics, using the convolution operation of the Gaussian function and the nature of the softenable kernel function, comprehensive use of the filtering method and multigrid method is proposed. The numerical instability is solved, and the filtering radius extension method is used to reduce the influence of local extreme values on the global optimisation of the structure [2]. The calculations show that the choice of the scheme is reasonable and effective, saving a lot of calculation time. At the same time, this paper chooses ANSYS software to realise the calculation of structural topology optimisation, and from the technical point of view, it realises the structural topology optimisation of hydraulic buildings.

Topology describes a special type of graphics, the so-called ‘topological properties’. However, although the topological property is a basic property of graphics, it also has strong geometric intuition; but it is difficult to accurately describe it in simple and popular language. Its exact definition is described in abstract language. Here we try to describe topology in terms of topological properties. Topological properties reflect the characteristics of the overall structure of the graphic. The graphic can be deformed (such as squeezed, stretched or twisted) at will; if it is not torn, no adhesion occurs, and the overall structure is not damaged. Nature remains the same. The above-mentioned deformations are called ‘topological transformations’ of the graphics, and then the topological properties are the properties that the geometric figures remain unchanged during the topological transformation. If the topology is described in terms of sets and mappings [3], topology is the study of the invariant or invariant properties of space under a topological transformation (or homeomorphism). The so-called homeomorphic space X and Y means that there is a two-way continuous (that is, reciprocal and continuous) correspondence between X and Y, which is figuratively that plasticine X can be kneaded into Y without allowing partitioning. Topological properties are geometric properties that are common to homeomorphic figures, so the two spaces X and Y of homeomorphic cannot be distinguished. Because the nature of topological research does not change when graphics are elastically deformed, topology is figuratively called ‘rubber geometry.’

The homogenisation method was proposed by Bendsoe and Kikuchi. It is the most widely used method for topology optimisation of continuum structures and belongs to the material description method. The basic idea is to introduce the microstructure (unit cell) into the material of the topological structure [4]. The form and size parameters of the microstructure determine the elastic properties and density of the micromaterial at this point. The optimisation process takes the size of the unit cell of the microstructure as topological design variables, the addition and deletion of microstructures are achieved by the growth and decline of the unit cell size, and composite materials composed of intermediate unit cells are generated to expand the design space and realise the unification and continuity of the structural topology optimisation model and the size optimisation model. The work on the homogenisation method mainly includes the research of the microstructure model theory and the practical application of the homogenisation model. The research on the theory of microstructure model mainly including proposed models of square guarding microstructure, two-level hierarchical microstructure, rectangular hollow microstructure, three-dimensional hierarchical microstructure, etc. pointed out that the orthogonal microstructure hypothesis will inevitably lead to incorrect results. Research on the application of the homogenisation model in the optimisation design of continuum topology. The research scope covers multi-load plane problems, 3D continuum problems, vibration problems, elasticity problems, buckling problems, 3D shell problems, thin shell structure problems, composite topology, optimisation issues and many other issues. The homogenisation theory is to divide the microstructure of many different holes in the design area (see Figure 1) to perform topology optimisation on the continuum.

In the process of homogenisation, it is difficult to keep all important physical quantities conserved before and after homogenisation. To prove this, consider the multigroup approximation of the neutron transport equation and assume that we can get the solution of the equation:
_{g}(_{g}(_{g}(_{g}(_{0} = Ω · Ω′, _{eff}_{g}(

After solving equation (1), the physical quantities that need to be conserved during the cell homogenisation process can be easily calculated. The three most important physical quantities are the average response rate of the grid element, the average interfacial flow and the eigenvalue (or proliferation coefficient). Like equation (1), the homogenisation equation can be obtained [5]:

The relationship between homogenisation parameters and non-uniform parameters is derived below according to the conservation of important physical quantities before and after homogenisation:

Due to the strong coupling between the homogenisation cross section and the homogenisation flux in the transport equation, it is not easy to calculate the homogenisation parameters to make equations (3) and (4) true. On the other hand, during the homogenisation process, a total of

Another feature of homogenisation is that it is sensitive to the external environment of the cell. It can be assumed that the core is the same for all cells, and there is no need to uniformise each cell at this time. However, in most realistic reactor problems, due to the different positions in the component, even the same cells will cause a difference in its external environment. People want to reduce the number of homogenisations of the same type of cells (basically, each type is performed only once), so the homogenisation parameters of a certain type of cells in a certain environment may not guarantee important physical quantities in the same cell in different environments.

This issue will be addressed in the proposed homogenisation technique.

The semi-homogenisation method guarantees the conservation of all-important physical quantities by introducing enough degrees of freedom. An iterative method is used to solve the homogenisation parameters.

Let _{αg}_{V} Σ_{αg}(_{g}(_{g}_{k} (

For one-dimensional problems with two surfaces, the homogenisation parameters can be determined by:

ANSYS program is a powerful and flexible large-scale general-purpose finite element commercial analysis software for design analysis and optimisation, integration of structure, heat, fluid, electromagnetic and acoustics. It is widely used in various industrial and scientific researches. The ANSYS program has basically realised seamless integration with CAD software: it has a strong and reliable meshing capability; the program has explicit integral calculations in the structure and multi-field analysis so that it can solve from linear problems to nonlinear problems and has the evolution from solving the structure field to solving the wobble field; at the same time, the ANSYS program has an open environment, and users can add their own tasks to the software to complete special tasks as needed [6].

The ANSYS program itself provides a direct use program for structural topology optimisation. At the same time, because of its powerful compatibility, if you want to use ANSYS to design a program for topology optimisation, you only need to combine it with the FORTRAN language for hybrid analysis. You can get the desired result. Because of the visual effects of ANSYS, the results of structural topology optimisation can be clearly seen, so the ANSYS program can be said to be a good implementation tool for structural topology optimisation.

The topology optimisation problem of a structure is equivalent to an inverse problem of a partial differential equation. The finite element method is a weak solution to the original problem on a finite-dimensional space _{h}

In fact, the grid dependency problem of the topology optimisation solution can be given a clear explanation by the characteristics of the integral equation solution. As can be seen from the previous section, the solutions to structural topology optimisation problems are ill-posed, but there are certain connections between the solutions. However, to find the numerical solution of the first type of Fredholm integral equation, the integral equation is first discretised to obtain:
_{j}_{j}_{i}_{i}_{j}_{i}_{j}_{i}_{j} f_{j}^{(1)} (_{j}^{(2)} (_{j}

Because the finite element method is an approximate method and the word length of a computer is limited, the calculated solution is only an approximate solution of a numerical model, that is, there is a certain error between the numerical solution and the physical understanding of the original problem. The so-called high-pass and low-pass terms of the error are relative to the smoothness of the numerical solution of the original problem. When the scale parameter of the function is constant relative to the selected grid size, the high-pass error components that can be eliminated for different initial grid functions are different. The multigrid method is an iterative method for solving large-scale equations and has been widely used in various disciplines and engineering and technical problems. The multigrid method can eliminate the low-pass part of the numerical solution error and accelerate the convergence of the solution to effectively solve the grid dependency problem in topology optimisation. On the premise of considering only static loads, according to the principle of virtual work in elasticity, the finite element equilibrium equation for plane stress problems is:
^{T}

Isoperimetric elements use two sets of coordinate systems, one is the actual coordinate system where the irregular quadrilateral elements are located, called the _{i}_{i}_{i}_{i}_{j}_{i}_{j}_{i}_{j}

The grid-independent filtering method is to modify the sensitivity information value of a unit by weighting the sensitivity information of neighbouring units in the surrounding neighbourhood of a specific unit to obtain the redistribution of the filtered unit sensitivity information as the next iteration.

Calculated initial value. Compared with other methods that try to eliminate the high-pass part of the grid-dependent numerical solution error, the grid-independent filtering method has little effect on the calculation time, and the algorithm is easy to implement and has been widely used in structural topology optimisation [8].

As mentioned earlier, the filtering method is to use different filter functions as the convolution kernel to convolve with the original digital signal. Here are some explanations on the selection of the convolution kernel (filter function). There are many classic filter functions in digital signals. In general, the selection of the filter function should follow the two principles: (1) The main lobe width of the frequency response of the filter function should be small, and the energy it contains should be as large as possible in the total energy. (2) The energy contained in the side lobe of the frequency response of the filter function varies with the bandwidth, which tends to decrease rapidly.

The above two principles are selected to consider the phenomenon of spectral leakage when the original function and the convolution kernel are convolved. Since the purpose of suppressing spectrum leakage can be achieved by selecting an appropriate filter function, how to choose an optimal filter function becomes the key. According to the Hegenss principle, the product of the time width

According to the formula of Granting Reservoir, the wave height is 2HL = 1.8 m, the wavelength is 2LL = 16.64 m and the average wave centreline is 0.85 m above the surface of the still water. The water depth in front of the dam is 95 m, which is greater than LL and is a deep-water wave. The calculated wave pressure per metre of single width is 109.9 kN. The load considers the basic load combination situation and considers upstream water pressure, downstream water pressure, lift pressure and gravity and wave pressure and does not consider sediment pressure and earth pressure. The topological optimisation of the structure is performed for the downstream water levels of 0 and 10 m. According to the basic requirements of the section design of the gravity dam, the height of the concrete gravity dam is initially set at 100 m. Because of the drainage pipe, the lifting pressure is effectively reduced, and so the width of the dam bottom is set to 75 m and the width of the dam top is 8 m [9, 10].

Set the basic design area as a 100 m × 75 m rectangular plane (see Figure 3), where W is the dead weight, P_{1} is the upstream water pressure, U is the lift pressure, and P_{L} is the wave pressure.

The grid size is divided into 1 m × 1 m, the concrete bulk density is 24 kN/m^{3} and the water bulk density is 10 kN/m^{3}. The downstream water level is 0. The width of the dam top is 8 m. In order to make the dam top have enough width, topology optimisation is forbidden at 0–8 m upstream to obtain a practical profile. The number of deleted element proportions is 55% of the optimised area. After the optimised results are obtained, the results are checked for sliding resistance and stress. To illustrate the problem, we do not take any measures to eliminate numerical instability in topology optimisation such as checkerboard format and grid dependency [11]. The parts of the dam crest width range are not subjected to topology optimisation. The results obtained are shown in Figure 4(a) and (b). Figure 4(a) shows the pseudo-density of the entire structure after topology optimisation, and the dashed box represents the original design area. Figure 4(b) shows the structure of the topology optimised in the form of a unit.

The thesis is closely related to the actual situation of hydraulic engineering. Based on the fractional partial differential equations and the theory of finite element analysis, the topological optimisation design of the non-overflow section of the concrete gravity dam, which is one of the most common forms of hydraulic engineering building structures, is proposed, a new type of dam section. The influence of the downstream water level pressure in the form of topology optimisation was compared. The results of the existing examples are used to compare the results of topology optimisation with commonly used optimisation results. The results prove the superiority of the topology optimisation idea.

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