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The importance of modern simulation methods in the mechanical analysis of heterogeneous solids is presented in detail. Thereby the problem is noted that even for small bodies the required high-resolution analysis reaches the limits of today’s computational power, in terms of memory demand as well as acceptable computational effort. A further problem is that frequently the accuracy of geometrical modeling of heterogeneous bodies is inadequate.

The present work introduces a systematic combination and adaption of grid-based methods for achieving an essentially higher resolution in the numerical analysis of heterogeneous solids. Grid-based methods are as well primely suited for developing efficient and numerically stable algorithms for flexible geometrical modeling. A key aspect is the uniform data management for a grid, which can be utilized to reduce the effort and complexity of almost all concerned methods. A new finite element program, called Mulgrido, was just developed to realize this concept consistently and to test the proposed methods. Several disadvantages which generally result from grid discretizations are selectively corrected by modified methods.

The present work is structured into a geometrical model, a mechanical model and a numerical model. The geometrical model includes digital image-based modeling and in particular several methods for the theory-based generation of inclusion-matrix models. Essential contributions refer to variable shape, size distribution, separation checks and placement procedures of inclusions. The mechanical model prepares the fundamentals of continuum mechanics, homogenization and damage modeling for the following numerical methods. The first topic of the numerical model introduces to a special version of B-spline finite elements. These finite elements are entirely variable in the order k of B-splines. For homogeneous bodies this means that the approximation quality can arbitrarily be scaled. In addition, the multiphase finite element concept in combination with transition zones along material interfaces yields a valuable solution for heterogeneous bodies. As the formulation is element-based, the storage of a global stiffness matrix is superseded such that the memory demand can essentially be reduced. This is possible in combination with iterative solver methods which represent the second topic of the numerical model. Here, the focus lies on multigrid methods where the number of required operations to solve a linear equation system only increases linearly with problem size. Moreover, for badly conditioned problems quite an essential improvement is achieved by preconditioning. The third part of the numerical model discusses certain aspects of damage simulation which are closely related to the proposed grid discretization. The strong efficiency of the linear analysis can be maintained for damage simulation. This is achieved by a damage-controlled sequentially linear iteration scheme.

Finally a study on the effective material behavior of heterogeneous bodies is presented. Especially the influence of inclusion shapes is examined. By means of altogether more than one hundred thousand random geometrical arrangements, the effective material behavior is statistically analyzed and assessed.

6 Download Section

6.1 First Exercise

Please note that Mulgrido is provided at a developer stage and has not been designed to cope with all kinds of possible user interaction at any time. In this context it is refered to Section 6.4. In the following it is described how to run a first example in Mulgrido.

The blue buttons are most important (Figure 7). In the beginning these buttons should only be pressed in the following order: Macro, Mesh, Meso, Loads and Solve. In each case a dialog window will open.

• Macro The geometry of an orthogonal body is defined. You can accept the given values and proceed with OK.
• Mesh A minimum number of degrees of freedom of the present mesh is defined. Again, you can just accept the given value and shut this dialog by OK.
• Show Model: Now you can enter the View menu and select option E-MODUL. You should see a green square.
• Meso Here, the material parameters of the model are defined and inclusions are generated by pressing Generate. If you like you can revisit Meso and repeat with different (useful) options. In each case the shown model should change.
• Loads For the first example the boundary conditions could be set as given in Figure 8 by the scroll down menus and Add. Correction is possible by Delete or Delete all. Leave this window by OK.

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  Figure 8: Possible valid boundary conditions

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• Solve For the first run use the predefined options, with Conjugate Gradient method and start the solution procedure by OK. Now you can enter the View menu and choose any option such as e.g. Stress Sxx. You might see a similar image as Figure 9. You can control the solution process by the buttons Pause, Cont., Step and Stop on the right. If you choose another solver method such as MG-Cycle please also Create a new cycle on the right of the solve window.

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  Figure 9: Snapshot during iteration procedure showing Stress Sxx.

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• View Controls: After or during the solution process you can modify the view controls, which are all placed on the right half of the menu. There you can define an isometric view such as that of Figure 10.

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  Figure 10: Isometric view of converged model showing Stress Sxx.

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Here ends the present description of this example. It is possible to use a few more options in this version of Mulgrido such as e.g. save and load models.

6.2 Second Exercise

This exercise demonstrates how to load image data into Mulgrido for the definition of an image-based heterogeneous model. The recognized image format is a 24-bit Windows Bitmap (R8 G8 B8) and the expected filename is ”bmpname.bmp”. Such an image can be downloaded here.

• Read Image The image is read from the same directory as the Mulgrido executable.
• Show Model: Enter the View menu and select option Material Type. Each color is considered as an individual materal. Switching the Color Palette to 9, shows the image model associated with its original colors, Figure 11.

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  Figure 11: Material type interpretation from loaded image data.

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• Meso For an interval of Material Number, a Young’s modulus and a Poisson’s ratio can be defined, using the corresponding Apply buttons, see Figure 12. Material properties need to be assigned to all material numbers of the model, in the present example 0-6. Material Info shows the settings in the text window. Leave the menu by Finish.

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  Figure 12: Menu to assign material properties to material type.

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• View Select E-Modul to show the defined distribution of Young’s modulus.
• Continue with Loads and Solve, analog to the first example.

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