Research on Topology Optimization for 3-D Body Type of
High Arch Dam
Chao SU 1 and Bei SUN 2 1College of Water Conservancy and Hydropower, Hohai University, P.O. Box 210098, Nanjing, P. R. China; PH (+86)2583787596; FAX (+86)2583731332; email: [email protected] 2Department of Port channels, Jiangsu Provincial Communication Planning and Design Institute Ltd., P.O. Box 210005, Nanjing, P. R. China; PH (+86)2583787596; FAX (+86)2583731332; email: [email protected]
ABSTRACT
Using the 3-D topology optimization theory and the feature of high arch dam to design the homologous program to build the 3-D model and design the body type for high arch dam with the fixed arch dam upstream face. The initiative result showed that the topology optimization can be used to the reasonable body type research for high arch dam.
Keywords: high arch dam; 3-D structure; topology optimization; homogenization method
1. BASIC THEORY AND CONDITION
The conventional arch dam body type optimization design is to optimize the parameters of prearranged body types for engineer application. The arch dam body type optimization is in the shape optimization stage, and the structural layout of arch dam is short of the supports from advanced system theory. The topology optimization belongs to the layout optimization and it is more difficult than other optimization methods. The topology optimization only needs material characters, optimization models and loads, and the other conventional optimizations need some parameter optimization variable such as objective function, status variable and design variable. In comparison with conventional optimization, the topology optimization is more challenging and has higher potentialities. The topology optimization with continuum initial configuration is paid more attention because of little dependency on the known knowledge of analysis object. The continuum structure optimization design with topology theory is the development tendency in structural optimization field now
The topology optimization with homogenization method disperses the design domain to discrete elements by the finite element method. Assume the discrete microstructure elements (unit cells) is homogeneous distribution and same size before optimization. The unit cell density distribution is changed in the course of the topology optimization: the unit cell density increases in the highly stressed areas and the unit cell density decreases in the lowly stressed areas. Then there is a supported structure in this process. A reasonable density minimum value is defined after the iterative computation completes completely, then some domains in which the density is lower than the minimum density value is deleted and the most optimal supported structure is obtained.
The physical dimension and azimuth of microstructure are design variables in the course of topology optimization calculation. The material density distribution of design field is determined by some optimization algorithm when some structure performance index reaches optimality, then the optimal solution will be determined. The criterion is selected to calculate in the course of topology optimization. Figure 1 showed the flowchart of structural topology optimization.
Figure 1. Flowchart of structural topology optimization
The mathematical model on homogenization method of topology optimization is:
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1. 2 SOLVENT ON NUMERICAL INSTABILITY PHENOMENA
There are some familiar numerical instability phenomena such as checkerboard pattern, mesh-dependence and local minimal phenomena. The topology optimization calculation can be equivalent to the inverse problem of partial differential equations. The checkerboard pattern as a numerical instability is easy to appear because of the ill-posedness on the solution of equation. Though the solutions are ill-posed, there are some links between these solutions, the numerical solution is not stable when using different ways of discretization of integral equation. The matrix conditions of linear algebraic equations will be more as the less of the finite size, so the solution of equation. It will bring the mesh-dependence to appear. At the same time, the local minimal phenomena will lead the achievement of globally optimal solution to be difficult usually.
The load conditions are very complex in arch dam. Although the immediate cause of numerical instability phenomena is not related to the structural loads and boundary conditions, the facts show the numerical instability phenomena are obvious with the more complex boundary condition and more kinds of load, and the topology optimization results will be infected more easily. In order to avoid the topology optimization results to be trashy, the numerical instability phenomena must be solved.
This article used a filtering method on the base of Gaussian function from the digital signal which had the character of the convolution operation and the constructable softening kernel function and the multi-grid method from the
hydraulics to solve the checkerboard pattern and mesh-dependence, and then extended radius of filter wave are introduced to reduce the influence of local minimal phenomena.
2. EXAMPLE ON ARCH DAM
2.1 BASIC CONDITION
A double-curvature arch dam, its height is 113m, the Upstream bears the pressure caused by the 109.2m water, while the downstream bears the Pressure caused by the 20m water. And the upstream and downstream water level drop to 89.2m, take the cartesian coordinate system as the coordinate system , the x-axis points from the left bank to the right bank ,the y-axis points to the upstream, the z-axis is vertical upward .And the dam body firstly build dam-Ring on the each floor, the upstream and downstream ,,then Bedrock of cross-strait are generated by the extrude of the both side of dam. At this case it can simulate roughly the terrain situation. Rock mass in the calculation of this paper will be considered as a homogeneous, Elastic modulus (EC) is 24GPa, Poisson's ratio( μc) is 0.167, Bulk density (γc) is 24KN/m, bedrock and foundation of cross-strait don’t consider defects such as the holes, so it is used the same parameters: Elastic modulus(EC) is 30GPa, Poisson's ratio(μc) is 0.25, Bulk density (γc) is 28.5KN/m.
The scope of the whole calculation of arch dam is from the top to the bottom, up to 113m, the width of the left and the right bank is 300m,and the upstream and downstream is also 300m.the bedrock is 250m from the dam bottom. Because the dam upstream face directly bear the load of water, if participating directly the topology optimization, it will be difficult to get the satisfied results which it is proved by many calculations. So they won’t participate the topology optimization. The upstream face of the cross-section curve arch dam uses the general conics, to conserve calculated space, dam body has the smaller grid, while the bedrock has the coarse grid. The basic structure is showed as Fig.2 and Fig.3。
The deletion ratio is 40%, the iteration step is 20.
2.2 RESULTS
Figure 4 shows the topology optimization result, and the saved finite unit density is 0.6 to 1.0. According to the figure 4, for arch dam in this example, from the downstream, the smaller stress units are removed gradually along the dam thickness direction in accordance with unit load, the tendency of leaf-by-leaf propulsion is very obvious. The whole arch dam is thinning gradually from bottom to top, and the downstream is a curved surface.
3. FITTING RESULTS
The topology optimization results are not used immediately in engineering project, it should be smooth fit appropriately, then it can be applied to practical projects.
Fit the arch dam downstream from figure 4, the fitting result is showed as figure 5(a). Figure 5(b) and (c) shows the fitting sketch map of arch ring in the height of 45m and the crown cantilever respectively.
The stress of fitting arch dam was be calculated and checked, the results are showed as figure 6(a) and figure 6(b). According to figure 6, the largest first principal stress value is 1.09MPa<1.2 MPa, the least third principal stress value is -5.00 MPa>-9.0 MPa, and the principal stress values all completely accord with standard requirement. There is stress concentration in dam heel, it should be caused take seriously, and some effective methods should be used to avoid the stress concentration.
5. CONCLUSIONS
The structural topology optimization can offer a conceptual design in initial stage in engineering structure design, it will obtain a better design on structure layout, and it has more generality for topology optimization compare with size optimization and shape optimization.
After reducing the conditions of batholite and load properly, the arch dam was calculated by homogenization method of topology optimization, the results showed the attempt has succeeded. The stresses of arch dam derived meet requirements. Because the loads and boundary conditions are all simplified, especially the temperature load is not considered, the practicality of topology optimization on arch dam need be further studied. The current study is focused on topology optimization in world structure optimum design field The study of topology optimization is just beginning in hydropower engineering project, and it has board prospect of research and application.
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Figure 6. Check result of stress (b) Third principal stress (a) First principal stress
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