Abstract

This paper intends to encourage a discussion on the geometrical and topological description of 3D computational models, and on the embedding of these models into the engineering design process. In the first part, we will report on the development of a mesh generator which tries to overcome some of the present problems of FEM meshers for thin-walled structures and is capable of generating hexahedral meshes for thin, curved geometries as they often appear in civil and mechanical engineering. The second part of the presentation will focus on the recently proposed Finite Cell Method (FCM), a fictitious domain approach, where even very complex geometries of the physical domain can be taken into account without any mesh generation. This new numerical method can be embedded in a computational steering environment for interactive computation, yielding an analysis and design environment, which is far more flexible and intuitive than classical finite elements in civil engineering.

Keywords: Geometric models, hexahedral mesh generation, thin walled structures, fictitious domain methods, Finite Cell Method, computational steering

1 Introduction This paper focuses on the geometrical and topological description of three-dimensional computational models, the necessity to generate conforming meshes and how to avoid them. 3D mesh generation for computational civil engineering is of large practical importance and still an issue of active research. Most of the total time spent for the entire design and analysis process in structural engineering is often devoted to the creation of a suitable geometry and the generation of a computational mesh. Only a fraction of the time is actually spent for the analysis itself. This ratio is even more unfavourable, if methods of high order are employed.

Whereas thin-walled structures have been modelled by dimensionally reduced elements (plates, shells) for decades, in recent years there is more and more demand for simulating these structures by thin solid elements, too. The reasons are obvious: First, for many nonlinear computations solid elements are much more adequate than their reduced counterparts, and many nonlinear formulations cannot easily be applied to shell elements. Second, coupling of thin-walled structures discretized by shell elements to massive parts modelled by solid elements often introduces severe modelling errors, and third, the ever increasing computational power relieves from the necessity to save computational time by using reduced formulations. Yet, the desire for real 3D-computations of thin-walled structures immediately raises the question of mesh-generation, in particular for models, which cannot be obtained by simply sweeping surface meshes to thin hexahedra. Although many 3D mesh-generators for tetrahedral meshes (Netgen) and some for pure hexahedral meshes (Cubit) are available, they

Geometric models, numerical analysis and computational steering in structural engineering

Ernst Rank, S. Kollmannsberger, D. Schillinger, Ch. Sorger, N. Zander Technische Universität München, Germany

cannot be applied to thin solid structures in many cases. These difficulties become even more pronounced, if high order elements in a p-version of the finite elements should be used. These are very favourable with respect to accuracy, robustness and numerical efficiency (Szabó et al., 2004), yet they have the disadvantage, that in case of shell modelling they need to be aligned to the exact surface of a (curved) structure. In the first part of this paper we will report on the development of a mesh generator that overcomes some of the present problems and is capable of generating hexahedral meshes for thin walled, curved geometries as they frequently appear in structural and civil engineering. We will present some geometrically elaborate examples and demonstrate the techniques with which corresponding, conforming meshes can be created for solid but shell like structures.

The second part of the paper will discuss the recently proposed Finite Cell Method (FCM), a fictitious domain approach, where even very complex geometries in two and three dimensions can be taken into account without any boundary fitting mesh generation. For this purpose, high order shape functions are spanned by a Cartesian grid that embeds the structure to be computed. Whereas fictitious domain methods (often also denoted as embedded domain or immersed boundary methods) have been studied in connection with low order elements for decades (e.g. Saul’ev, 1963), their extension to high order methods has only been proposed recently (Düster et al., 2008; Rank et al., 2011). It turns out, that this combination yields an astonishingly powerful simulation technique, reducing the engineering effort for preparing the analysis model drastically, and, at the same time, gives an accuracy and efficiency, which can in many cases not be achieved by classical finite element methods. Examples for two and three dimensional structural problems will demonstrate the potential and the large practical advantages of this method compared to mesh based procedures.

In a final section it will be shown how the Finite Cell Method can be integrated in a Computational

Steering system allowing an interactive structural simulation even for geometrically complex three-dimensional structures.

2 Hexahedral mesh generation for thin-walled structures

2.1 Single Surface meshing

Our hexahedral mesh generator (for a detailed description see (Sorger et al., 2012) starts with the generation of a quadrilateral reference mesh, which can e.g. be defined on the middle plane of the thin solid structure. In addition to the reference mesh, an upper and a lower surface of the thin solid needs to be defined (Figure 1 left). We use a domain splitting procedure in a parameter plane as described in (Schweingruber, 1999) combined with a mapping onto the surface to obtain a first quadrilateral mesh of the reference surface (Figure 1 middle). This mesh is swept along the normal of the reference surface and defines together with the upper and lower bounding surface the hexahedral mesh of the thin solid (Figure 1 right).

Figure 1: Left: Reference surface, upper and lower surface of thin solid Middle: Quadrilateral mesh on reference surface Right: Thin solid mesh

2.2 Multiple Surface meshing

A multiple shell solid model is composed of a set of the previous section. The presented algorithm combines the solid models depending on the topological connectivity of their reference surfaces. Figure 2 shows three examples of multiple shell solid models.

Figure 2: Intersections of multiple shell models

Of major importance for multiple shell modelFigure 3).

Figure 3: a) Reference surfaces b) Offset model c) Offset model with quadrilateral meshskeleton e) Hexahedral mesh

The mesh generation starts from a

composed of four segments and a covering plane. In a first step an originally connected surfaces are offset from there intersection curves, so that unconnected surface patches are obtained. These can be meshed into quadrilateral surface elements and extruded to hexahedra. An important aspect of thwhich are meshed such that a conforming interface skeleton consisting of quadrilaterals and hexahedra can be generated. This interface skeleton is connected to the extruded hexahedral meand results in the full hex-mesh of the thin

A multiple shell solid model is composed of a set of intersecting single shell solid modelthe previous section. The presented algorithm combines the solid models depending on the topological connectivity of their reference surfaces. Figure 2 shows three examples of multiple shell

multiple shell models

rtance for multiple shell modelling is the identification of the interface skeleton

Figure 3: a) Reference surfaces b) Offset model c) Offset model with quadrilateral meshes d) Interface

from a geometric model of the reference surfaces, here a cylinder composed of four segments and a covering plane. In a first step an offset model is generated, where all originally connected surfaces are offset from there intersection curves, so that unconnected surface patches are obtained. These can be meshed into quadrilateral surface elements and extruded to hexahedra. An important aspect of this step is a mutual referencing of the corresponding offset curves, which are meshed such that a conforming interface skeleton consisting of quadrilaterals and hexahedra can be generated. This interface skeleton is connected to the extruded hexahedral me

mesh of the thin-walled structure.

single shell solid models as shown in the previous section. The presented algorithm combines the solid models depending on the topological connectivity of their reference surfaces. Figure 2 shows three examples of multiple shell

interface skeleton (see

d) Interface

here a cylinder is generated, where all

originally connected surfaces are offset from there intersection curves, so that unconnected surface patches are obtained. These can be meshed into quadrilateral surface elements and extruded to

is step is a mutual referencing of the corresponding offset curves, which are meshed such that a conforming interface skeleton consisting of quadrilaterals and hexahedra can be generated. This interface skeleton is connected to the extruded hexahedral meshes

2.3 An complex example

Figure 4, left shows the geometric model of a wind turbine defined by B

(Rhino) as a geometric modeller. Figure 4, rightgeometry of the whole structure. In Figure 5be noted that not only nodes lie on the true surfaces, but that all elements themselves are aligned to the spline surfaces. This is necessary if have the important advantage that large elements with high as

Figure 4: Left: Geometric model of a wind turbine Right:

Figure 5: Details of an all-hex mesh for high order elements

3 The Finite Cell Method

3.1 The basic principles

Whereas we have shown in the last section how high order finite element meshes, consisting of curved hexahedral meshes only can be generated, we will now demonstrate that these elements are powerful enough to eliminate the necessity of geometry aligned meshing at all.Finite Cell Method (for details, Parvizian et al.Ruess et al., 2011, Rank et al., 2011domain Ω (Figure 6 (a)), up to the boundary of a fictitious domain

rectangular and can be easily meshed with a simple, regular Cartesian grid

shows the geometric model of a wind turbine defined by B-splines Figure 4, right depicts the interface skeleton. Figure 5, leftIn Figure 5, details of the all-hex-meshes are shown, where it should

be noted that not only nodes lie on the true surfaces, but that all elements themselves are aligned to s. This is necessary if high order elements are applied (Düster et al.

have the important advantage that large elements with high aspect ratios can be used.

etric model of a wind turbine Right: (Part of ) interface skeleton

hex mesh for high order elements

The Finite Cell Method

Whereas we have shown in the last section how high order finite element meshes, consisting of meshes only can be generated, we will now demonstrate that these elements are

powerful enough to eliminate the necessity of geometry aligned meshing at all. The basic idea of this Parvizian et al., 2007, Düster et al., 2008, Schillinger et al.

2011) is to extend the partial differential equation beyond the physical , up to the boundary of a fictitious domain CΩ∂ (Figure 6 (b,c

rectangular and can be easily meshed with a simple, regular Cartesian grid (Figur

splines using Rhino leton. Figure 5, left shows the

meshes are shown, where it should be noted that not only nodes lie on the true surfaces, but that all elements themselves are aligned to

(Düster et al., 2001), which

Whereas we have shown in the last section how high order finite element meshes, consisting of meshes only can be generated, we will now demonstrate that these elements are

The basic idea of this Schillinger et al., 2011,

is to extend the partial differential equation beyond the physical (Figure 6 (b,c)) which is

(Figure 6 (d)). To

differentiate from the standard elements, these rectangular grid elements are called shape functions are employed to approximate the displacement field in each cell, while the strain and stress fields are approximated with their first

Figure 6: The basic concept of the Finite Cell Method

In course of the finite cell computation (which is, up to the integration of element matrices, identical to a high order finite element computation), iby introducing a penalty parameter problem. It takes the value 1 inside the physical domain, whereas positive value for reasons of numerical stability) outside the physical domain. the domain Ω are treated as usual finite elements of high order, cells lying completely outside are neglected. Only cells cut by the boundary

Figure 7: Octree for numerical integration of cell matrix

A composed numerical integration scheme being based on an octree decomposition of the cell is used. Thus, the discontinuity can be integrated been investigated for problems of linear elasticity in twoDüster et al., 2008), geometrically and physically nonlinear problems (Schillinger et al., 2011), to problems from biomechanics (Yang well as to multiphysics and multiscale problems FCM has been extended to thin-walled structures example from thermoelasticity and one for thin solid shells will be discuss

(a) (b) (c)

differentiate from the standard elements, these rectangular grid elements are called cells

pproximate the displacement field in each cell, while the strain and stress fields are approximated with their first-order derivatives.

Figure 6: The basic concept of the Finite Cell Method

In course of the finite cell computation (which is, up to the integration of element matrices, identical to a high order finite element computation), it is distinguished between physical and fictitious domain by introducing a penalty parameter α being used as a factor to the material tensor of the structural

inside the physical domain, whereas it is set to 0 (or to a very small positive value for reasons of numerical stability) outside the physical domain. Whereas cells inside

are treated as usual finite elements of high order, cells lying completely outside are Only cells cut by the boundary Ω∂ need special attendance see Figure 7.

Figure 7: Octree for numerical integration of cell matrix

A composed numerical integration scheme being based on an octree decomposition of the cell is used. Thus, the discontinuity can be integrated with sufficient accuracy. This finite cell approach has been investigated for problems of linear elasticity in two and three dimensions (Parvizian et al., 20Düster et al., 2008), geometrically and physically nonlinear problems (Abedian

), to problems from biomechanics (Yang et al., 2011, Ruess et al., 2011physics and multiscale problems (Zander et al, 2012; Sehlhorst, 2011). Recently, the

walled structures (Rank et al., 2011) modelled by solid cells. example from thermoelasticity and one for thin solid shells will be discussed in the next sections.

(b) (c)

cells. High-order pproximate the displacement field in each cell, while the strain and

In course of the finite cell computation (which is, up to the integration of element matrices, identical t is distinguished between physical and fictitious domain used as a factor to the material tensor of the structural

(or to a very small Whereas cells inside

are treated as usual finite elements of high order, cells lying completely outside are

A composed numerical integration scheme being based on an octree decomposition of the cell is ell approach has

and three dimensions (Parvizian et al., 2007, Abedian et al, 2010,

, Ruess et al., 2011), as ). Recently, the

modelled by solid cells. An ed in the next sections.

(d)

3.2 A thermoelastic example

Figure 8: Heat exchanger – finite element versus finite cell results

Consider a heat exchanger as outlined in Figure 8, together with a finite element mesh (left) and

the finite cell grid (right). A coupled problem is investigated with a first solution of the heat equation followed by an elastic analysis of the structure. Finite element results (left) are compared to the finite cell analysis (right, polynomial degree of the shape functions used p=10), where none of the cell boundaries corresponds to boundaries of the domain. Temperature distribution as well as thermal stresses throughout the structure coincide very well, demonstrating that high accuracy is obtained by the FCM without the necessity to generate a boundary fitting mesh. More details of this example together with a convergence analysis are given in (Zander et al., 2012).

In the next example, a modification of the well-known Scordelis Lo benchmark problem (e.g. Scordelis and Lo, 1969) is shown. Details of the structure are depicted in Figure 9, left. In contrast to the original benchmark a hole is punched into the shell. We compare again results of the Finite Cell Method with those of a reference solution obtained with an overkill solution. Whereas the finite cell grid completely neglects the hole in the structure, the finite element computation uses a mesh which accurately follows the structural boundary. Figure 9, right, depicts convergence in energy norm of FCM and a p-FEM-extension. Both methods converge with exponential preasymptotic rate and show an accuracy, which would not be practically achievable with low order finite elements. FCM is, comparing degrees of freedom, even more accurate as p-FEM.

Figure 9: Left: Modified Scordelis Lo shell Right: Convergence in energy norm

In Figure 10 von Mises stresses are compared, showing the finite element results together with the mesh of thin solid elements on the left, whereas the finite cell results with the grid of computation are depicted on the right. Note again that cell boundaries do not coincide with the boundary of the circular hole in the structure.

Figure 10: Von Mises stress for modified Scordelis Lo shell. Left: Reference solution Right: FCM result

4 Computational Steering

In this section we demonstrate the computational efficiency of the Finite Cell Method and show

that it can even be used for interactive numerical simulation computational steering system.

Computational Steering, as defined by (Mulder et al.,

of the parameters of the simulation process during its execution and presumes the availability of a front-end to analyze the effects of these interactions immediately.

A common approach in computational

different communicating processes. They may then be optimized for performance on suitable hardware. We closely follow this approach.implemented in C under Linux and compiled on a standard workstation (Dell Precision T5500, 24GB Memory, 2 x Intel Xeon CPU 3.33GHz). The visualization kernel implemented in Visual C++ under Windows and runs on a different workstation wigraphics card (NVDIA Quadro FX 5800 graphics card with 4GB memory hosted in a Windows 7 machine with 4GB RAM and 2 x Intel Xeon CPU 2.67GHz). This general setting is depicted in Figu11 .

Figure 11: Structure of the Computational Steer

The communication between the two machines is realized through a standard TCP socket connection with a capacity of one Gbps using a client server model. load 3D CAD models of a structure, define boundary conditions simulation systems to modify boundary conditions and even the After the initial setup only the changevisualization kernel on the one and the simulation kernel on the other side.depends on the fact, that even a change of only the outcome of an inside-outside

Computational Steering using the Finite Cell Method

e the computational efficiency of the Finite Cell Method and show that it can even be used for interactive numerical simulation (Yang et al., 2012), i.e. in a so

fined by (Mulder et al., 1999), enables to directly change some or all of the parameters of the simulation process during its execution and presumes the availability of a

end to analyze the effects of these interactions immediately.

A common approach in computational steering is to separate the visualization and simulation into different communicating processes. They may then be optimized for performance on suitable hardware. We closely follow this approach. The Finite Cell Method in the simulation kernel was

ed in C under Linux and compiled on a standard workstation (Dell Precision T5500, 24GB Memory, 2 x Intel Xeon CPU 3.33GHz). The visualization kernel (see Dick et al., 2011implemented in Visual C++ under Windows and runs on a different workstation wigraphics card (NVDIA Quadro FX 5800 graphics card with 4GB memory hosted in a Windows 7 machine with 4GB RAM and 2 x Intel Xeon CPU 2.67GHz). This general setting is depicted in Figu

Structure of the Computational Steering system

communication between the two machines is realized through a standard TCP socket connection with a capacity of one Gbps using a client server model. Our steering system allows to load 3D CAD models of a structure, define boundary conditions and loads and, in contrast to standard

modify boundary conditions and even the topology of the structure After the initial setup only the changes in the model are communicated between user interface and

el on the one and the simulation kernel on the other side. This system strongly fact, that even a change of topology does not require any remeshing of a structure,

outside-test of integration points need to be communicated

e the computational efficiency of the Finite Cell Method and show ), i.e. in a so-called

, enables to directly change some or all of the parameters of the simulation process during its execution and presumes the availability of a

steering is to separate the visualization and simulation into different communicating processes. They may then be optimized for performance on suitable

simulation kernel was ed in C under Linux and compiled on a standard workstation (Dell Precision T5500, 24GB

(see Dick et al., 2011) was implemented in Visual C++ under Windows and runs on a different workstation with a powerful graphics card (NVDIA Quadro FX 5800 graphics card with 4GB memory hosted in a Windows 7 machine with 4GB RAM and 2 x Intel Xeon CPU 2.67GHz). This general setting is depicted in Figure

communication between the two machines is realized through a standard TCP socket Our steering system allows to

and loads and, in contrast to standard of the structure on the fly.

s in the model are communicated between user interface and This system strongly

does not require any remeshing of a structure, communicated.

We explicitly emphasize that no high-end supercomputer needs to be used as a back end of our

simulation system. Instead, standard desktop hardware at a current price of approximately 3000 Euros provides sufficient computational power to produce direct feedback in form of realtime visualization at an update rate of about one picture/second.

5 Conclusions

Even after 70 years of computational structural simulation in civil engineering a lot of improvement in computer supported structural design and analysis is possible and necessary. It is still a huge practical challenge to integrate geometric modeling and numerical simulation, and still a lot of engineering time has to be devoted to the transition between these models. In this paper we have shown that mesh generators which take explicit advantage of special topological features of a structure and which incorporate NURBS-based shapes rather than only classical facet models provide significant improvements, especially if thin solid computation instead of dimensionally reduced models are desired. Furthermore, we have introduced a new simulation technique, the Finite Cell Method, which completely relieves from any necessity to mesh a structure. Despite the high accuracy which is achievable by this method it can be implemented in a very efficient way. Embedded in a computational steering framework, the Finite Cell Method thus opens the door for a new paradigm in structural simulation: Instead of the classical separation into the steps preprocessing, computation, postprocessing it is now possible to design software systems following the principle of interactive

numerical simulation for structural problems.

Acknowledgements

The authors gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (DFG) under grant RA 624/19-1, the International Graduate School of Science and Engineering and the Munich Centre of Advanced Computing at TUM.

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