A Graph Grammar Model of the hp Adaptive Three

Fundamenta Informaticae 114 (2012) 183–201

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DOI 10.3233/FI-2012-623 IOS Press

A Graph Grammar Model of the hp Adaptive Three Dimensional Finite Element Method. Part II. ´ Anna Paszynska, Ewa Grabska Faculty of Physics, Astronomy and Applied Computer Science Jagiellonian University, Reymonta 4, 30-059, Krakow, Poland [email protected]; [email protected]

´ ∗ Maciej Paszynski AGH University of Science and Technology, Krakow, Poland [email protected]

Abstract. This paper presents a composite programmable graph grammar model of the three dimensional self-adaptive hp Finite Element Method (hp-FEM) algorithms. The computational mesh composed of hexahedral finite elements is represented by a composite graph. The operations performed over the mesh are expressed by composite graph grammar productions. The three dimensional model is based on the extension of the two dimensional model for rectangular finite elements. This paper is concluded with numerical examples, presenting the generation of the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL), the modern patterning process. Keywords: Graph grammar, Automatic hp adaptivity, Finite Element Method, Step-and-Flash Imprint Lithography

1. Introduction This paper presents a composite programmable graph grammar model for the self-adaptive three dimensional hp Finite Element Method algorithm (3D hp-FEM). The 3D self-adaptive hp-FEM is the most ∗

Address for correspondence: AGH, University of Science and Technology, Al.Mickiewicza 30, 30-059, Kraków, Poland

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A.Paszyńska, E. Grabska, M. Paszyński / Graph Grammar Model of hp-Adaptive 3D FEM

sophisticated version of adaptive algorithm, providing the exponential convergence of the numerical error with respect to the mesh size [5], for a class of elliptic and Maxwell problems. In this paper, Composite Programmable Graph Grammars (CP-GG) are used to model the mesh transformations which result from the execution of the 3D self-adaptive hp-FEM algorithm. The 3D computational mesh is represented by an appropriate composite graph. The first part of this paper deals with the 2D model consisting of rectangular and triangular elements, while the second part of the paper describes the 3D model for hexahedral meshes, based on the 2D model introduced in the first part. This is the first attempt to express three dimensional hp-adaptive mesh transformations by means of the graph grammar model. The presentation is concluded with a numerical simulation presenting the execution of the introduced graph grammar transformations on the cube shape 3D mesh. The execution is driven by the 3D selfadaptive hp-FEM algorithm [5], utilized to simulate the Step-and-Flash Imprint Lithography, a modern patterning process [1, 4, 2].

2. Graph transformations for modelling the 3D hp-adaptive Finite Element Method This section presents the CP-graph grammar suitable for modelling transformations of three dimensional meshes with hexahedral elements. The composite graph modelling a mesh is defined with the following sets of graph node labels AV : • A1V = {ICB, ICE, IC, IC1, C} is a set of node labels which denote hexahedral elements, • A2V = {vv, vv2} is a set of node labels which denote nodes of hexahedral elements, • A3V = {F F, EE, F F i, EEe, F F e} is a set of node labels which denote edges of hexahedral elements, • A4V = {F A, F F A} is a set of node labels which denote faces of hexahedral elements, • A5V = {II, JJ, Ii, JJ3, JJ4, ii} is a set of node labels which denote interiors of faces of elements, • A6V = {ln, in} is a set of node labels which denote interiors of elements. • A7V = {S, B, F AL, Z, f al} is a set of node labels which denote additional nonterminals. The set of all grammar productions is divided into four subsets: • the set of grammar productions responsible for generation of the structure of the initial mesh, which consists of nodes which correspond to types of mesh elements, • the set of grammar productions generating the structure of hexahedral elements, • the set of grammar productions for the identification of common faces of two adjacent hexahedral elements,


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• the set of grammar productions for breaking hexahedral elements. The first three subsets of productions are responsible for the generation of the topology of the mesh. The last set consists of productions which model h-refinement (breaking element into son elements).

2.1. CP-graph grammar productions responsible for the generation of the structure of the initial 3D mesh This set of productions generates the linear structure of the initial mesh. The left and right-hand-side of each production consist of nodes with labels corresponding to element types (see: Figure 1). The label IC1 denotes the hexahedral element. Additionally, the labels corresponding to the first, the last and the single element of the mesh have to be introduced, because of the different number of bonds. The nodes with label ICB, ICE and IC denote the first, the last and the single hexahedral element of the mesh, respectively. The node with label S is a start symbol (the axiom of the grammar). The node with label Z is an additional nonterminal. This set of productions can be used to generate the structure of a mesh which consists only of hexahedral elements. Figure 2 presents an exemplary execution of a sequence of productions which generate the structure of an initial mesh.

Figure 1. A set of grammar productions responsible for the generation of the structure of an initial mesh.

Figure 2. Generation of the structure of a 3D initial mesh.


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2.2. Grammar productions generating the structure of hexahedral elements After the generation of the initial mesh structure, the structure of each element must be generated. Figure 3 presents the production for generating the graph structure of an hexahedral element.

Figure 3.

A grammar production (P Generate 3D) which generates the structure of a hexahedral element.

Similar productions are defined for nodes labelled by ICB, ICE and IC1.

2.3. Grammar productions for identification of common faces of two adjacent hexahedral elements In this subsection the production which identifies a common face of two adjacent hexahedral elements is presented. The production is similar to the production, which identifies common edge of two adjacent 2D elements. Figure 4 presents the production for identifying common faces of two adjacent hexahedral elements.


Figure 4. A graph grammar production (P Identify 3D Face) for the identification of common faces.

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2.4. Grammar productions for breaking elements Once the structure of a hexahedral finite elements is generated, we can proceed with mesh refinements. The h-refinement consists in breaking selected elements into eight son elements, in order to increase accuracy of approximation in that area.The refinement procedure is expressed by breaking element faces, edges and interiors. The process of breaking an element consists of following steps: • breaking elements interiors, • allowing for breaking interiors of faces, if interiors of two adjacent elements are broken, • breaking a face, • allowing for breaking an edge, if the interiors of appropriate faces are broken, • breaking an edge, • allowing for breaking elements interiors. Breaking elements interiors. Breaking an element interior means generating 8 new element interiors, 12 new faces, 6 new edges and 1 new node, as it is shown in Fig. 5. The new faces are colored in light grey, new edges in dark grey and the new node in black color.

Figure 5.

A hexahedral element broken into 8 new interiors, 12 new faces, 6 new edges and 1 new node.

The production breaking the interior of a hexahedral element is presented in Fig. 6. The new created element interiors are nodes with label Jn, nodes with label JJ denote new face interiors, nodes with label F F A denote new faces, nodes with label F F i denote new edges and nodes with label V V denote the new nodes. Some edges in the composite graph from the right hand-side of


Figure 6.

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Graph grammar production (P Break 3D Interior) for breaking an interior of a hexahedral element.

the production are omitted in order to make the figure more readable. Fig. 7 presents in details such fragment of this composite graph that corresponds to one of the newly created elements: the right upper back hexahedral element. Allowing for breaking interiors of faces. Interiors of faces can be broken, if interiors of two adjacent elements are broken or a face is adjacent to one broken interior and to the boundary of the domain. Fig. 8 presents the productions which allow for breaking an interior of a face, if this condition is fulfilled. These productions change the node label which represents an interior of a face from II to Ii. Breaking an interior of a face of an hexahedral element. The production of Fig. 9, which represents the operation of breaking a face interior can be used only for nodes with label Ii. Allowing for breaking an edge. An element edge can be broken if the interiors of appropriate adjacent faces are broken. Fig. 10 presents the productions which allow for breaking an edge, if this condition is fulfilled. These productions change the label of the node which represents an edge from F F to EE. Breaking an edge. The production of Fig. 11, which represents the operation of breaking an edge, can be used only for nodes with label EE. Allowing for breaking elements interiors. After breaking an edge, an hexahedral element interior can be broken if it is not adjacent to a large unbroken edge. The production is similar to production for rectangular elements in Fig. 41 of the first part of the paper. It checks whether a hexahedral element interior is adjacent to proper size edges in 3D. In this case the number of adjacent proper size edges is recorded by changing the graph node symbol denoting the element interior from JJ into JJ3 and finally into JJ4.

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Figure 7. Fragment of the composite graph from the right hand-side of the production for breaking the interior of a hexahedral element.

Figure 8. The productions (P Allow Break Face) and (P Allow Break Boundary Face) which allows for breaking a face interior of a hexahedral element.


Figure 9.

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The production (P Break Face) for breaking the interior of a face of a hexahedral element.

Figure 10. The productions (P Allow Break Edge) and (P Allow Break Boundary Edge) which allow for breaking the edge of a hexahedral element.

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Figure 11. The production (P Break Edge) for breaking the edge of a hexahedral element.

3. Numerical example 3.1. Step-and-Flash Imprint Lithography The example presented in this section illustrate the proposed model and it is related to Step and Flash Imprint Lithography (SFIL). It is a patterning process utilizing photopolymerization to replicate the topography of a template onto a substrate [1, 4, 2, 14]. The nanolithography methods like the SFIL are utilized for the production of micro-processors. The SFIL process can be described in the following six steps, as it is illustrated in Figure 12. 1. Dispense. The SFIL process employs a template / substrate alignment scheme to bring a rigid template and substrate into parallelism, trapping the etch barrier in the relief structure of the template, 2. Imprint. The gap is closed until the force that ensures a thin base layer is reached, 3. Exposure. The template is then illuminated through the backside to cure etch barrier, 4. Separate. The template is withdrawn, leaving low-aspect ratio, high resolution features in the etch barrier, 5. Breakthrough etch. The residual etch barrier (base layer) is etched away with a short halogen plasma etch, 6. Transfer etch. The pattern is transferred into the transfer layer with an anisotropic oxygen reactive ion etch, creating high-aspect ratio, high resolution features in the organic transfer layer. The major processing steps of SFIL include: depositing a low viscosity, silicon containing, photocurable etch barrier on to a substrate; bringing the template into contact with the etch barrier; curing the etch barrier solution through UV exposure; releasing the template, while leaving high-resolution features behind; a short, halogen break-through etch; and finally an anisotropic oxygen reactive ion etch to yield high aspect ratio, high resolution features. Photopolymerization, however, is often accompanied by densification. Densification of the SFIL photopolymer (the etch barrier) may affect both the cross sectional shape of the feature and the placement of relief patterns. The exemplary shrinkage of the feature measured after removing the template is presented in Figure 13. The linear elasticity model with thermal expansion coefficient is used to verify the material response of polymerized networks in cured etch-barrier layers that are formed during the exposure, and after removal of the template.


Figure 12. The SFIL process.

Figure 13. The shrinkage of the feature measured after removing the template.

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We focus on the simulation of the deformation of the feature after removal of the template. It is assumed that the polymer network has been damaged during the removal of the template, and thus the interparticle forces are weaker in one part of the mesh. The problem has been solved on the 3D cube shape domain, presented in Figure 14.

Figure 14. The initial 3D mesh.

3.2. Linear elasticity model with thermal expansion coefficient Following [10] the strong and weak formulations for the linear elasticity problem with thermal expansion coefficient are given as follows. Strong formulation. Given gi : ΓD ∋ x → gi (x) ∈ R, θ and αkl , find the displacement vector field ¯ ∋ x → ui (x) ∈ R, i = 1, 2, 3, such that ui : Ω σij,j = 0 in Ω,

(1)

ui = gi on ΓD ,

(2)

where σij is the stress tensor, defined in terms of the generalized Hook’s law σij = cijkl (ǫkl + θαkl ) ,

(3)

here cijkl are elastic coefficients (known for given material), θ is the temperature, αkl are the termal u +u expansion coefficients, and ǫij = u(i,j) = i,j 2 j,i is the strain tensor, where ui,j are displacement gradients. Weak formulation. The weak formulation is obtained by multiplying (1) by test functions wi ∈ H01 (Ω) and integrating by parts over Ω: Z Z − wi,j σij dΩ + wi σij nj dΩ = 0. (4) Ω

Γ


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Since σij is symmetric tensor, then wi,j σij = w(i,j) σij and Z

Ω

w(i,j) σij dΩ = 0.

(5)

where we have also used the fact that wi = 0 on Γ. Finally, by utilizing (3) we get Z Z w(i,j) cijkl u(k,l) dΩ = −θ w(i,j) cijkl αkl dΩ. Ω

(6)

Ω

Reformulation for the SFIL modeling. For the convenient implementation of the algorithm, we utilize the following equivalent weak formulation. Find u ∈ V, such that a (u, w) = −A (w) ∀w ∈ V, a (u, w) =

Z

(7)

ǫ (w)T Dǫ (u) dΩ

(8)

ǫ (w)T DαdΩ

(9)

Ω

A (w) = θ

Z

Ω

3 where V = {V ∈ H 1 (Ω) : trv = 0 on ΓD }, and ΓD is defined as the bottom of the 3D cube. Here 

     ǫ(u) =      

u1,1 u2,2 u3,3 u2,3 + u3,2 u1,3 + u3,1 u1,2 + u2,1



     ,    

1−ν ν ν   ν 1−ν ν   ν 1−ν E  ν D=   (1 + ν)(1 − 2ν) 0 0  0  0 0 0  0 0 0

(10)

0 0 0 1−2ν 2

0 0

0 0 0 0 1−2ν 2

0 0 0 0 0

0

1−2ν 2



     .    

(11)

In the following simulation, we assume that Young modulus E = 1 GPa and Poisson ration ν = 0.3, as provided by [3]. We also assume that gi : ΓD ∋ x → gi (x) = 0 (the feature is fixed at the bottom, with free boundary conditions on all other sides), θ = 1 (the thermal expansion coefficient α expresses the volumetric contraction of the feature when the temperature gradient is equal to 1 Celsius), αij = −αδij where α = −0.0615 is based on inverse analysis [13]. The material damage that is supposed to result is slight lean of the feature, has been modeled by decreasing the value of the Young modulus E = 0.001 GPa on one half of the feature, from 20 to 40 percent of its height.

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Discussion on numerical results. The self-adaptive hp-FEM algorithm has generated a sequence of meshes delivering exponential convergence of the numerical error with respect to the number of degrees of freedom. The initial mesh, presented in Figure 14 can be obtained by executing the following sequences of graph grammar productions (P4)-(P Generate 3D)-(P Break 3D Interior)-(P Allow Break Boundary Faces)6 -(P Break Face)6 -(P Allow Break Edge)12 -(P Break Edge)12 At this point, we have an initial mesh element broken into 8 smaller son elements, with all its 6 faces located on the boundary broken, as well as all its 12 edges located on the boundary, also broken. Now we break each of the 8 son elements, first by breaking their interiors, then by breaking all 24 faces located on the boundary, as well as breaking all 12 internal faces. Finally, we break all 72 edges located on the boundary as well as we break all 36 internal edges. This is expressed by the following sequence (P Break 3D Interior)8 -(P Allow Break Boundary Faces)24 -(P Break Face)24 -(P Allow Break Faces)12 -(P Break Face)12 - (P Allow Break Boundary Edge)72 -(P Break Boundary Edge)72 -(P Allow Break Edge)36 -(P Break Edge)36 . The presented sequences result in the mesh presented in Figure 14. At this point the computational problem is solved by utilizing a multi-frontal solver, that can be also modeled as graph grammar productions [15]. Finally, we perform error estimations over all son elements from the mesh, and we select the refinement for 33 percent of elements with highest numerical error. We execute selected refinements over these elements. The elements can be either p refined, which corresponds to assigning different polynomial orders of approximations for finite element edges, faces and interiors, or h refined, which corresponds to executions of (P Break 3D Interior) graph grammar productions. The p refinement is expressed by attributing composite graph nodes representing element vertices, edges, faces and interiors with polynomial orders of approximation. The different polynomial orders of approximations are illustrated in Figures 15-18 by using different colors. Notice that an element can have different polynomial orders of approximation over its edges and over its faces in both directions. These polynomial orders of approximation are expressed in Figures 14, 15, 16 and 17 by different colors assigned to element edges and faces. For the case of a face, the horizontal order of approximation over a face can be read from the color over the edge in the horizontal direction, while the vertical order of approximation over a face can be checked from the color over the edge in the vertical direction. In the case that a face of a just broken element is located on the boundary, the productions -(P Allow Break Boundary Faces)-(P Break Face) are fired. In the case that two adjacent finite elements has been broken, the common face is broken by executing (P Allow Break Faces)-(P Break Face). In the case where an edge of the just broken element is located on the boundary, the productions (P Allow Break Boundary Edge)-(P Break Boundary Edge) are executed. Finally, if all faces sharing an edge have been broken, the edge is broken by executing (P Allow Break Edge)-(P Break Edge). The mesh transformations required by the algorithm have been expressed as executions of graph grammar productions over the composite graph model for the three dimensional initial mesh, presented in Figure 14. The meshes generated after the first, second and third iterations of the algorithm are presented in Figures 15, 16 and 17 respectively. Finally, the x, y and z components of the solution - displacement vector field - are presented in Figures 18, 19 and 20, respectively. The assumed material damage, modelled by decreasing the value of the Young modulus, resulted in slight lean of the feature in the direction of the damage, which is especially visible in Figure 18, presenting the x component of the displacement vector field.


Figure 15. The optimal mesh obtained after the first iteration of the self-adaptive hp-FEM.

Figure 16. The optimal mesh obtained after the second iteration of the self-adaptive hp-FEM.

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Figure 17. The optimal mesh obtained after the third iteration of the self-adaptive hp-FEM.

Figure 18. The x component of the displacement vector field. The minimum displacement denoted by blue color corresponds to -32 nanometers, the maximum displacement denoted by red color corresponds to 65 nanometers.


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Figure 19. The y component of the displacement vector field. The minimum displacement denoted by blue color corresponds to -44 nanometers, the maximum displacement denoted by red color corresponds to 44 nanometers.

Figure 20. The z component of the displacement vector field. The minimum displacement denoted by blue color corresponds to -90 nanometers, the maximum displacement denoted by red color corresponds to 0 nanometers.

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4. Conclusions and future work This paper proposes the composite programmable graph grammar model of the three dimensional selfadaptive hp Finite Element Method (hp-FEM) algorithm. The three dimensional mesh with hexahedral finite elements was represented by a composite graph. The mesh transformation performed by the 3D self-adaptive hp-FEM algorithm were expressed by graph grammar productions. The three dimensional model was obtained by extension of the 2D model for rectangular and triangular finite elements. The model was verified by executing the graph grammar productions over the three dimensional cube shape mesh. The graph grammar productions were selected by the 3D self-adaptive hp-FEM algorithm in order to generate the optimal mesh for simulation of the Step-and-Flash Imprint Lithography (SFIL), a modern patterning process. Future work will involve incorporation of the tetrahedral elements into the 3D model. The main technical difficulty is that the tetrahedral element breaks into several tetrahedral elements and two pyramid elements, which makes the transformations very complicated, see [5]. The next step will involve the expression of the 3D multi-frontal direct solver algorithm by the graph grammar model. Having the complete 3D graph grammar model for both mesh transformations and solver execution, we will perform the Partitioning Communication Aglomeration and Mapping (PCAM) analysis as proposed by [6]. The analysis has been already performed for the two dimensional hp-FEM with rectangular finite elements only [11, 12]. We will also analyze the concurrency of the algorithm on the level of atomic tasks, tasks and super tasks, as it has been already done for the two-dimensional algorithm [15]. It is also possible to employ the Petri nets model for detections of deadlocks, infinite executions or starvation in the process of mesh transformations, similarly to the two dimensional analysis already presented in [16, 17].

5. Acknowledgements The work presented in this paper has been partially supported by Ministry of Scientific Research and High Education grants no. NN 519 405737 and NN 501 120836.

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