Advanced Mesh Update Techniques for Problems Involving Large Displacements

Transcription

1 WCCM V Fifth World Congress on Computational Mechanics July 7,, Vienna, Austria Eds.: H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner Advanced Mesh Update Techniques for Problems Involving Large Displacements Keith Stein Department of Physics, Bethel College St. Paul, MN, 55, USA k-stein@bethel.edu Tayfun Tezduyar Mechanical Engineering, Rice University MS 3 6 Main Street, Houston, Texas 775, USA tezduyar@rice.edu Key words: mesh moving, fluid-structure interactions, solid-extension mesh moving technique Abstract The mesh update methods we use in computation of fluid-structure interactions are based on meshmoving and remeshing-as-needed. For problems with complex geometries and large structural displacements, it is a high priority to design mesh moving techniques that reduce the need for remeshing. For that purpose, we present in this paper mesh moving techniques where the motion of the nodes is governed by the equations of elasticity, with selective treatment of mesh deformation based on element sizes as well as deformation modes in terms of shape and volume changes. We also present here how the Solid- Extension Mesh Moving Technique (SEMMT) functions as part of our mesh update method. We present application of these approaches to a set of D test cases.

2 Keith Stein, Tayfun Tezduyar Introduction The Deforming Spatial Domain/Stabilized Space Time (DSD/SST) formulation [,, 3] was developed for computation of flows with moving boundaries and interfaces, including fluid-structure interactions. The DSD/SST formulation is an interface-tracking technique, and as such requires that the mesh be updated to track the moving interfaces as the spatial domain occupied by the fluid varies (i.e. deforms) with respect to time. In computations with the Arbitrary Lagrangian-Eulerian method, which is another interface-tracking method, one faces the same requirement. In general, mesh update consists of moving the mesh for as long as it is possible, and full or partial remeshing (i.e., generating a new set of elements, and sometimes also a new set of nodes) when the element distortion becomes too high. As the mesh moves, the normal velocity of the mesh at the interface has to match the normal velocity of the fluid. With this condition met, our main objective in designing a mesh update technique becomes reducing the remeshing frequency. This is very important in 3D computations with complex geometries, because remeshing in such cases typically requires calling an automatic mesh generator and projecting the solution from the old mesh to the new one. Both of these steps involve large computational costs. In most practical problems, such as the parachute fluid-structure interactions, the complexity of the overall problem geometry would require that the mesh be produced with an automatic (unstructured) mesh generator, and such a mesh would require an automatic mesh moving technique. In the automatic mesh moving technique introduced in [4], the motion of the nodes is governed by the equations of elasticity, and the mesh deformation is dealt with selectively based on the sizes of the elements and also the deformation modes in terms of shape and volume changes. The motion of the internal nodes is determined by solving these additional equations. As boundary condition, the motion of the nodes at the interfaces is specified to match the normal velocity of the fluid at the interface. Mesh moving techniques with comparable features were introduced in [5]. In the technique introduced in [4], selective treatment of the mesh deformation based on shape and volume changes is attained by adjusting the relative values of the Lamé constants of the elasticity equations. The objective would be to stiffen the mesh against shape changes more than we stiffen it against volume changes. Selective treatment based on element sizes, on the other hand, is attained by altering the way we account for the Jacobian of the transformation from the element domain to the physical domain. In this case, the objective is to stiffen the smaller elements, which are typically placed near solid surfaces, more than the larger ones. The method described in [4] was recently augmented in [6] to a more extensive kind, by introducing a stiffening power that determines the degree by which the smaller elements are rendered stiffer than the larger ones. When the stiffening power is., the method reduces back to an elasticity model with no Jacobian-based stiffening. When it is., the method is identical to the one introduced in [4]. In our studies we investigate the optimum values of this stiffening power with the objective of reducing the deformation of the smaller elements. In this context, by varying the stiffening power, we generate a family of mesh moving techniques, and test these techniques on fluid meshes where the structure undergoes three different types of prescribed motion or deformation. In the mesh moving technique introduced in [4], the structured layers of elements generated around solid objects (to fully control the mesh resolution near solid objects and have more accurate representation of the boundary layers) move glued to these solid objects, undergoing a rigid-body motion. No equations are solved for the motion of the nodes in these layers, because these nodal motions are not governed by

3 WCCM V, July 7,, Vienna, Austria the equations of elasticity. This results in some cost reduction. But more importantly, the user has full control of the mesh resolution in these layers. For early examples of automatic mesh moving combined with structured layers of elements undergoing rigid-body motion with solid objects, see [7]. Earlier examples of element layers undergoing rigid-body motion, in combination with deforming structured meshes, can be found in []. In computation of flows with fluid-solid interfaces where the solid is deforming, the motion of the fluid mesh near the interface cannot be represented by a rigid-body motion. Depending on the deformation mode of the solid, we may have to use the automatic mesh moving technique described above. In such cases, presence of very thin fluid elements near the solid surface becomes a challenge for the automatic mesh moving technique. In the Solid-Extension Mesh Moving Technique (SEMMT) introduced in [8, 9], we proposed treating those very thin fluid elements almost like an extension of the solid elements. In the SEMMT, in solving the equations of elasticity governing the motion of the fluid nodes, we assign higher rigidity to these thin elements compared to the other fluid elements. Two ways of accomplishing this were proposed in [8, 9]; solving the elasticity equations for the nodes connected to the thin elements separate from the elasticity equations for the other nodes, or together. If we solve them separately, for the thin elements, as boundary conditions at the interface with the other elements, we would use traction-free conditions. In this paper, we demonstrate how the SEMMT functions as part of our mesh update method. We employ both of the SEMMT options described above. Here, we will refer to the separate solution option as SEMMT Multiple Domain and the unified solution option as SEMMT Single Domain. For the SEMMT Single Domain option, we assign higher rigidity to the thin (inner) elements by setting the stiffening power to. for those elements while leaving it at the standard value of. for the other (outer) elements. Mesh Moving Model. Equations of Linear Elasticity Let IR n sd be the spatial domain bounded by,, wherensd is the number of space dimensions. Corresponding to the Dirichlet- and Neumann-type boundary conditions, the boundary, is composed of, g and, h. The equations governing the displacement of the internal nodes can then be written as r + f = on ; () where is the Cauchy stress tensor and f is the external force. For linear elasticity, is defined as = tr("(y))i + "(y); () where y is the displacement, tr() is the trace operator, and are the Lamé constants, I is the identity tensor, and "(y) is the strain tensor: "(y) = The Dirichlet- and Neumann-type boundary conditions are represented as ry +(ry) T : (3) y = g on, g ; n = h on, h : (4)

4 Keith Stein, Tayfun Tezduyar. Finite Element Formulation In writing the finite element formulation for Eq. (), we first define the finite element trial and test function spaces S h and V h : S h = fy h jy h [H h ()] n sd ; y h : = g h on,g g; (5) V h = fw h jw h [H h ()] n sd ; w h : = on,g g: (6) Here, H h () is the finite-dimensional function space over. The finite element formulation for Eq. () is then written as follows: find y h S h such that 8w h V h Z "(w h ):(y h )d, Z w h f d = Z, h w h hd,: (7) By assigning appropriate values to the ratio =, we can produce to a certain extent the desired effect in terms of volume and shape changes for the elements during the mesh motion. This approach becomes more clear if we re-write the term that generates the stiffness matrix as where "(w h ):(y h )=( + n sd ) tr("(w h )) tr("(y h ))+" (w h ):" (y h ); (8) " (y h )="(y h ), n sd tr("(y h ))I: (9) The two terms on the right-hand-side of Eq. (8) can be recognized as those corresponding, respectively, to the volume and shape change components of the stiffness matrix. In this context, the relative values of ( + n ) sd and can be adjusted to produce to a certain extent the desired effect in terms of stiffening the mesh against volume or shape changes. Although a selective treatment of the mesh deformation can be incorporated also into the force vector f by providing an appropriate definition for the forcing function, in our case we set it equal to zero..3 Jacobian-Based Stiffening A selective treatment of the mesh deformation based on the element sizes can be attained by altering the way we account for the Jacobian of the transformation from the element domain to the physical domain. This method was first introduced in [4], where the Jacobian is dropped from the finite element formulation, resulting in the smaller elements being stiffened more than the larger ones. Here we describe how that method is augmented to a more extensive kind. For this purpose we first write the global integrals generated by the terms in Eq. (8) as Z XZ [:::]d = [:::] e J e d; () e where [...] symbolically represents what is being integrated, is the finite element (parent) domain, and the Jacobian for element e is defined as J e =det(@x=@) e, with x and representing the physical and element (local) coordinates.

5 WCCM V, July 7,, Vienna, Austria We alter the way we account for the Jacobian as follows: Z [:::] e J e d 7,! Z [:::] e J e J J e d; () where, a non-negative number, is the stiffening power, and J, an arbitrary scaling parameter, is inserted into the formulation to make the alteration dimensionally consistent. With =:, the method reduces back to an elasticity model with no Jacobian-based stiffening. With =:, the method is identical to the one first introduced in [4]. In the general case of 6= :, the method stiffens each element by a factor of (J e ),,and determines the degree by which the smaller elements are rendered stiffer than the larger ones..4 Solid-Extension Mesh Moving Technique (SEMMT) In the SEMMT Multiple Domain option, we solve the elasticity equations for the nodes connected to the thin (inner) elements separately from the elasticity equations for the other nodes. As boundary conditions at the interface with the other elements, we use traction-free conditions. In the SEMMT Single Domain option, we solve the elasticity equations for all the nodes together, but stiffen the inner elements to a greater extent. We accomplish that in this paper by setting = : for those elements while leaving it at the standard value of =: for the outer elements. 3 Test Cases The test cases are all based on a D unstructured mesh consisting of triangular elements and an embedded structure with zero thickness. The mesh spans a region of jxj : and jyj :. The structure spans y =: and jxj :5. In the test cases for the Jacobian-based stiffening, a thin layer of elements (with `y =:) are placed along each side of the structure, with 5 element edges along the structure (i.e. `x = :). Figure shows the mesh and its close up view near the structure. In tests cases for the SEMMT, three of such thin layers of elements are placed along each side of the structure. 3. General Test Conditions and Mesh Quality Measures The test cases involve three different types of prescribed motion or deformation for the structure: rigidbody translation in the y-direction, rigid-body rotation about the origin, and prescribed bending. In the case of prescribed bending, the structure deforms from a line to a circular arc, with no stretch in the structure and no net vertical or horizontal displacement. In all test cases the maximum displacement or deformation is reached over 5 increments. Those maximum values are y = :5 for the translationtest, a rotation of = =4 for the rotation test, and bending to a half circle ( = ) for the bending test. The mesh over which the elasticity equations are solved is updated at each increment. This update is based on the displacements calculated over the current mesh that has been selectively stiffened. That way, the element Jacobians used in stiffening are updated every time the mesh deforms. As a result, the most current size of an element is used in determining how much it is stiffened. Also as a result, as an element approaches a tangled state, its Jacobian approaches zero, and its stiffening becomes very large. To evaluate the effectiveness of different mesh moving techniques, two measures of mesh quality are defined based on those used in []. They are Element Area Change (fa e ) and Element Shape Change

6 Keith Stein, Tayfun Tezduyar Figure : D test mesh for Jacobian-based stiffening. (f e AR ): f e A = log A e A e o =log (:) ; f e AR = log AR e AR e o =log (:) : () Here subscript o refers to the undeformed mesh (i.e., the mesh obtained after the last remesh) and AR e is the element aspect ratio, defined as AR e = (`emax) A e ; (3) where `emax is the maximum edge length for element e. For a given mesh, global area and shape changes (f A and f AR ) are defined to be the maximum values of the element area and shape changes, respectively. 3. Test Results for Jacobian-Based Stiffening In these tests, ranges from : (no stiffening)to :. In the translation tests, the prescribed translation is in the y-direction, with the displacement magnitudes ranging from y =:5 to :5. Figure shows the deformed mesh for the maximum translationof y =:5. It is evident that the small elements near the structure respond poorly for =:, resulting in severe stretching of the row of elements adjacent to the structure, and tangling of elements near the structure tips. For =: and =:, thesmall elements near the structure experience no tangling and significantly less deformation. For =:, the small elements near the structure undergo almost rigid-body motion. However, the behavior of the larger elements deteriorates as the smaller elements are stiffened. This is most apparent for =: where the larger element tangle near the upper boundary of the mesh. Figure 3 shows the values of f A and f AR as functions of and for different magnitudes of translation. The bold curve crossing the contours denotes the value of that results in minimum global mesh deformation. For example, for a displacement of :5 the optimal value of f A is obtained when is approximately :5. For larger displacements, the optimal value of is slightly greater. The optimal value of f AR is obtained at :8 for a displacement of :5 and at :7 for a displacement of :5.

7 WCCM V, July 7,, Vienna, Austria Figure : Jacobian-based stiffening translation tests. Deformed mesh for =:; :; : y = y = f A f AR χ Area Change χ Shape Change Figure 3: Jacobian-based stiffening translation tests. Mesh quality as function of stiffening power.

8 Keith Stein, Tayfun Tezduyar Figure 4: Jacobian-based stiffening rotation tests. Deformed mesh for =:; :; :. In the rotation tests, the rotation magnitudes range from = :5 to :5. For = : the mesh experiences significant stretching and tangling near the structure tips. No tangling is seen for the cases with element stiffening, but for = : the large elements near the outer boundaries experience significant distortion. Figure 4 shows the deformed mesh for the maximum rotation of =4. Figure 5 shows the values of f A and f AR as functions of and for different magnitudes of rotation. The minimum deformation of the mesh is seen for values of around :8. The mesh qualitydeteriorates more rapidly as decreases from : than when increases from :. In the bending tests, the bending magnitudes range from =: to, where denotes the arc length (in radians) for the deformed structure. Figure 6 shows the deformed mesh when the structure bends to a half circle (i.e. = ). For =:, we see tangling near the structure tips. As the element stiffening increases, tangling at the tips disappears, but severe element distortion arises in the interiors. Figure 7 shows the values of f A and f AR as functions of and for different magnitudes of prescribed bending. The minimum deformation of the mesh is seen for values of around :. 3.3 Test Results for the SEMMT These tests are carried out with the standard technique (where =: for all the elements), SEMMT Single Domain (with = : for the inner elements and = : for the outer elements), and SEMMT Multiple Domain (with =: for all elements in both domains). Figure 8 shows, for SEMMT Multiple Domain, the deformed mesh for the translation, rotation, and bending tests. Figure 9 shows, in close-up views, comparison of these meshes with the deformed meshes

9 WCCM V, July 7,, Vienna, Austria θ =.5 π.5 π 4.5 θ =.5 π.5 π f A f AR χ Area Change χ Shape Change Figure 5: Jacobian-based stiffening rotation tests. Mesh quality as function of stiffening power. Figure 6: Jacobian-based stiffening bending tests. Deformed mesh for =:; :; :.

10 Keith Stein, Tayfun Tezduyar θ =. π. π 4.5 θ =. π. π f A f AR χ Area Change χ Shape Change Figure 7: Jacobian-based stiffening bending tests. Mesh quality as function of stiffening power. obtained with the standard technique and SEMMT Single Domain. Figure shows, for the translation test, the two mesh quality measures (defined based on the inner elements) for the standard technique, SEMMT Single Domain, and SEMMT Multiple Domain. For the mesh quality measures defined based on all the elements, there is little difference between the three mesh moving techniques. Figure shows, for the rotation test, the two mesh quality measures (defined based on the inner elements) for the standard technique, SEMMT Single Domain, and SEMMT Multiple Domain. For the mesh quality measures defined based on all the elements, there is little difference between the three mesh moving techniques. Figure shows, for the bending test, the two mesh quality measures (defined based on the inner elements) for the standard technique, SEMMT Single Domain, and SEMMT Multiple Domain. Figure 3 shows, for the bending test, the two mesh quality measures (defined based on all the elements) for the standard technique, SEMMT Single Domain, and SEMMT Multiple Domain. 4 Concluding Remarks We have presented automatic mesh moving techniques for fluid-structure interactions with large displacements. In these techniques, the motion of the nodes is governed by the equations of elasticity, and deformation of the elements are treated selectively based on element sizes as well as deformation modes in terms of shape and volume changes. Smaller elements, typically placed near solid surfaces, are stiffened more than the larger ones. This is attained by altering the way we account for the Jacobian of the transformation from the element domain to the physical domain. The degree by which the smaller elements are stiffened more than the larger ones is determined by a stiffening power introduced into the formulation. When the stiffening power is :, the method reduces back to a model with no Jacobianbased stiffening. The D test cases we presented here for three different structural deformation modes show that the stiffening power approach substantially improves the deformed mesh quality near the solid surfaces, even when the displacements are large. The test cases also show that the optimal stiffening power is somewhat problem-dependent. It is higher for the bending tests ( :) than it is for the rotation ( :8) and translation ( :7)tests. We also presented how the Solid-Extension Mesh Moving Technique (SEMMT) functions as part of

11 WCCM V, July 7,, Vienna, Austria Translation Rotation Bending Inner Mesh Full Mesh Close-up View Figure 8: Mesh deformation tests with SEMMT Multiple Domain.

12 Keith Stein, Tayfun Tezduyar Translation Rotation Bending SEMMT SD SEMMT MD Figure 9: Mesh deformation tests with the standard mesh moving technique, SEMMT Single Domain, and SEMMT Multiple Domain.

13 WCCM V, July 7,, Vienna, Austria f A. f AR y y Figure : Translation test mesh quality (defined based on the inner elements) as function of translation magnitude, for the standard mesh moving technique, SEMMT SD, and SEMMT MD f A. f AR θ/π θ/π Figure : Rotation test mesh quality (defined based on the inner elements) as function of rotation magnitude, for the standard mesh moving technique, SEMMT SD, and SEMMT MD f A. f AR θ/π θ/π Figure : Bending test mesh quality (defined based on the inner elements) as function of bending magnitude, for the standard mesh moving technique, SEMMT SD, and SEMMT MD.

14 Keith Stein, Tayfun Tezduyar f A f AR θ/π θ/π Figure 3: Bending test mesh quality (defined based on all the elements) as function of bending magnitude, for the standard mesh moving technique, SEMMT SD, and SEMMT MD. our mesh update method. We employed two options of SEMMT. In SEMMT Multiple Domain, the elasticity equations for the nodes connected to the thin (inner) elements are solved separately from the elasticity equations for the outer nodes. For the inner elements, as boundary conditions at the interface with the outer elements, we use traction-free conditions. In SEMMT Single Domain, the elasticity equations for the inner and outer nodes are solved together, but we assign higher rigidity to the inner elements. The test computations for translation, rotation, and bending tests show that the SEMMT results in significant improvements in mesh quality for the inner elements. Acknowledgments This work was supported by the Army Natick Soldier Center and NASA JSC. References [] T. Tezduyar, Stabilized finite element formulations for incompressible flow computations, Advances in Applied Mechanics, 8, (99), 44. [] T. Tezduyar, M. Behr, J. Liou, A new strategy for finite element computations involving moving boundaries and interfaces the deforming-spatial-domain/space time procedure: I. The concept and the preliminary tests, Computer Methods in Applied Mechanics and Engineering, 94(3), (99), [3] T. Tezduyar, M. Behr, S. Mittal, J. Liou, A new strategy for finite element computations involving moving boundaries and interfaces the deforming-spatial-domain/space time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders, Computer Methods in Applied Mechanics and Engineering, 94(3), (99), [4] T. Tezduyar, M. Behr, S. Mittal, A. Johnson, Computation of unsteady incompressible flows with the finite element methods space time formulations, iterative strategies and massively parallel implementations, in P. Smolinski, W. Liu, G. Hulbert, K. Tamma, eds., New Methods in Transient Analysis, AMD-Vol.43, ASME, New York (99), pp. 7 4.

15 WCCM V, July 7,, Vienna, Austria [5] A. Masud, T. Hughes, A space time Galerkin/least-squares finite element formulation of the Navier-Stokes equations for moving domain problems, Computer Methods in Applied Mechanics and Engineering, 46, (997), 9 6. [6] K. Stein, T. Tezduyar, R. Benney, Mesh moving techniques for fluid structure interactions with large displacements (), to appear in Journal of Applied Mechanics. [7] T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, S. Mittal, Parallel finite-element computation of 3D flows, IEEE Computer, 6(), (993), [8] T. Tezduyar, Finite element interface-tracking and interface-capturing techniques for flows with moving boundaries and interfaces, inproceedings of the ASME Symposium on Fluid-Physics and Heat Transfer for Macro- and Micro-Scale Gas-Liquid and Phase-Change Flows (CD-ROM), ASME Paper IMECE/HTD-46, ASME, New York, New York (). [9] T. Tezduyar, Stabilized finite element formulations and interface-tracking and interface-capturing techniques for incompressibleflows (), to appear in Proceedings of the Workshop on Numerical Simulations of Incompressible Flows, Half Moon Bay, California. [] A. Johnson, T. Tezduyar, Simulation of multiple spheres falling in a liquid-filled tube, Computer Methods in Applied Mechanics and Engineering, 34, (996),