List of Figures

• Top: Unstructured prism mesh, Bottom: p-type convergence for the Helmholtz problem with $\lambda=1$ and solution sin(y)sin(z). The face geometry was downloaded from the web-site: http://www.3dcafe.com/
• Hybrid discretization around a multi-element airfoil. Only part of the domain is shown.
• Hierarchy of the code
• The tensor coordinates of the triangle.
• The tensor coordinates of the quadrilateral.
• The tensor coordinates of the tetrahedron.
• The tensor coordinates of the pyramid.
• The tensor coordinates of the prism.
• The tensor coordinates of the hexahedron.
• Diagram of the local top and base vertex of the tetrahedron.
• Setting up the required connectivity for the discretization of a box as shown in a). Vertex A is given as the first local top vertex as shown in b). In c) vertex B is then given as the local base vertex and the local base vertices from group one are aligned to satisfy connectivity and the final element orientation is shown.
• Connectivity summary. Solid arrows imply we can connect elements together with no extra mesh refinement, dashed arrows mean that there are some constraints on allowable configurations.
• Mode shapes for the triangle modal basis (${\bf TrHH}$) with N=5.
• Top: Geometric ordering for modes, Bottom: Stiffness matrix $(\nabla \phi_i, \nabla \phi_j)$ for the continuous basis for triangles (N=15) with the geometric ordering of the rows and columns.
• Top: Polynomial degree ordering for modes, Bottom: Stiffness matrix $(\nabla \phi_i, \nabla \phi_j)$ for the continuous basis for triangles (N=15) with modes ordered by polynomial order.
• Mode shapes for the quadrilateral modal basis (${\bf QuHH}$) with N=5.
• Mode shapes for the quadrilateral mixed modal basis ${\bf QuHN}$ with N=5.
• Mass matrix for the quadrilateral mixed modal basis (${\bf QuHN}$) with N=15.
• Mode shapes for the quadrilateral nodal basis (${\bf QuNN}$) with N=5.
• Mode shapes for the triangle mixed basis (${\bf TrNH}$) with N=5.
• Comparison of the elemental mass matrix for (a) ${\bf TrHH}$, (b) ${\bf TrNH}$ using collocation, and (c) ${\bf TrNH}$ using exact integration for N=15.
• Comparison of the elemental stiffness matrix for (a) ${\bf TrHH}$, and (b) ${\bf TrNH}$ bases for N=15.
• A piece of code demonstrating how a list of elements can be constructed.
• A piece of code demonstrating how operations can be made on a list of elements.
• Meshes (A-H) are consisted of three quadrilaterals and two triangles which are progressively skewed by shifting the interior vertex.
• Convergence in the L₂ norm for modal projection of the function $u=sin(\pi x)sin(\pi y)$ on meshes A-H.
• Convergence in the L₂ norm for mixed projection of the function $u=sin(\pi x)sin(\pi y)$ on meshes A-H.
• Illustration of local and global numbering for a domain containing one quadrilateral and one triangular elements. Here the expansion order is N=3, and we only show the boundary modes.
• Z-matrix map from global to local degrees of freedom
• Exponential accuracy is achieved for the wave equation with $u=sin(\pi cos(\pi x))$ as initial condition.
• Spectral radius of the weak convective operator on a periodic domain, N=12.
• Growth of the spectral radius of the Galerkin convective operator with expansion order.
•  $\rho_\Box(\theta)$ is an exact fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with regular quadrilaterals and with N=12.
•  $\rho_\triangle(\theta)$ is a close fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with triangles and with N=12.
• Spectral radius of the Galerkin convective operator on a periodic domain discretized with non-regular elements.
• Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six tetrahedra and fourth-order expansion.
• Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six pyramids and fourth-order expansion.
• Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with two prisms and fourth-order expansion.
• Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with eight hexahedra and fourth-order expansion.
• Form of the global operator matrix. Notice the sparsity of A and the block nature of B and C.
• Here we show the domain that we solved the Helmholtz equation in the following forcing function, $f = -(3.4 \pi^{2})\sin{(\pi.x)} cos{(\pi y)} \cos{(0.2 \pi.z)}$ and boundary conditions $u = \sin{(\pi.x)} cos{(\pi.y)} \cos{(0.2 \pi.z)}$, with periodic boundary conditions at the spiral ends. We see a dramatic bandwidth reduction because of the aspect ratio of the mesh and the exponential decay in the error with increasing expansion order.
• The exponential decay in the error with increasing expansion order for the Helmholtz equation solved on the spiral domain.
• Convergence test for the Helmholtz equation using quadrilaterals and triangles, with Dirichlet boundary conditions. The exact solution is $u=sin(\pi x)cos(\pi y)$ and forcing function $f=-(\lambda + 2 \pi^2)sin(\pi x)cos(\pi y)$.
• Convergence test for the Helmholtz, using a triangle and a quadrilateral, with Dirichlet boundary conditions: u=sin(pi cos(pi r**2)).
• Convergence for the Helmholtz problem on a mesh of hybrid elements. Dirichlet boundary conditions u=sin(x)sin(y)sin(z) and lambda = 10000.
• Convergence is guaranteed for the Helmholtz problem even on very skewed tetrahedra. Dirichlet boundary conditions $u=sin(\frac{\pi x}{6})sin(\frac{\pi y}{6})sin(\frac{\pi z}{6})$ and $\lambda=1$.
• Discretized solution domain for the Wannier-Stokes flow using 68 elements. Left: Graph of exponential spatial convergence, Top Right: Hybrid spectral element mesh, Bottom Right: Streamlines of the steady state Stokes flow.
• Steady state solution for the Kovasznay flow at Reynolds number Re=40 using a discretization using K=24 elements. Left: The spectral element mesh used, Right: Steady state streamlines.
• Convergence in the L infty and H1 norms as a function of expansion order for the steady state Kovasznay flow at a Reynolds number Re=40.
• Top Left: Three-dimensional spectral element mesh used to solve Kovasznay flow on, Top Right: Iso-contours of streamwise component of velocity
• Bottom: Convergence to exact solution with increasing order.
• Domain for the two-dimensional incompressible flow past a circular cylinder simulations.
• Mesh for the two-dimensional incompressible flow past a circular cylinder simulations.
• Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re=50,100 .
• Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re= 150.
• Instantaneous contours of vorticity for incompressible flow past a two-dimensional cylinder at Reynolds numbers Re=200 and 250.
• Top: The faces of the tetrahedra at the nose of the fish. Bottom: The spline surface on the body of the Robotuna.
• Robotuna simulation. Re=1000 based on height of tuna. N=5. Iso-contours of pressure are shown. Flow is from left to right.
• Top: Spectral element mesh KTri=50. Lower: Time convergence history for zero initial conditions N=5. Lower Right: Convergence to exact solution with increasing N.
• Lower Right: Convergence to exact solution with increasing N.
• Sketch of the micro-pump operating between two micro-channel systems. Inlet and exit valves open and close periodicly with maximum gap of gmax=0.125L and minimum gap of gmin=0.025L.
• Top: Deflection of the membrane y(x,t)=a sin(pi x/L) sin(omega t) Bottom: Loci of the valve tips y(t)= +-tanh(4 cos(omega t))
• Spectral element mesh used for the discretization of the micro-pump system at ejection-stage (top, membrane is moving up). The discretization of the flow domain during suction stage is shown at the bottom (membrane is moving down). The bottom figure also shows elemental discretization obtained by 7th order modal expansions used.
• Non-dimensional volumetric flow rate variation with in a period of the micro-pump, as a function of Reynolds number, Re=a*a*omega/nu, a/L = 1/10, a/h=1/3.
• Vorticity contours for Re=30 simulation. Top figure at tau omega = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified. Middle figure is at tau omega =0.72, corresponding to the end of the suction stage. A vortex jet pair is visible in the pump cavity. Bottom figure tau omega = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.
• Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified.
• Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega =0.72, corresponding to the end of the suction stage. A Vortex jet pair is visible in the pump cavity.
• Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.
• Three-dimensional micropump simulation Re=3. Top: An instant during the suction stage. Bottom: An instant during the ejection stage.
• Incompressible magnetic Pearson's vortex (t=2, instantaneous fields). Top: Periodic spectral element mesh, Bottom Left: Pressure field, Bottom Right: Magnetic streamfunction.
• Incompressible magnetic Pearson's vortex. Convergence plot for L2 error in x-component of magnetic field at time t=2 versus expansion order.
• Incompressible magnetic Pearson's vortex. Time accuracy plot for the simulation run with N=10. Convergence plot for L2 error in x-component of velocity at time t=1 versus time step Delta t.
• Incompressible Orszag-Tang vortex (t=1, instantaneous fields). Top Left: x component of velocity, Top Right: y component of velocity, Middle: Pressure, Bottom Left: x component of magnetic field, Bottom Right: y component of magnetic field.
• Incompressible Orszag-Tang vortex (t=1, instantaneous fields). Top Left: Velocity streamlines, Top Right: Magnetic streamlines, Middle Left: Vorticity, Middle right: Divergence of velocity, Bottom Left: Curl of magnetic field, Bottom Right: Divergence of magnetic field.
• Incompressible flow past a cylinder with inflow magnetic fields. From the top: (1) x component of the velocity field, (2) y component of the velocity field, (3) x component of the magnetic field, (4) y component of the magnetic field
• Incompressible flow past a cylinder with inflow magnetic fields aligned with inflow velocity. Top: Pressure field, Middle: Vorticity, Lower: Stream function of the magnetic field
• Density contours obtained on a hybrid grid for an inviscid M=0.3 flow (Top), on a triangle grid (Middle). The bottom plot shows exponential convergence of the error for the unstructured (triangles) and the hybrid (squares) grid.
• Inviscid M=0.3 flow past a bump in a three-dimensional (periodic in the spanwise direction) domain. From the top: (1) domain, (2) spectral element mesh used in the convergence test K=120, (3) Iso-Contours of density and (4) convergence of entropy to zero with increasing order.
• Instantaneous iso-contours for the simulation of compressible flow past a cylinder (Re=100, M=0.5) From the top: (1) density, (2) pressure field, (3) x component of the velocity field, (4) y component of the velocity field
• Instantaneous iso-contours of vorticity for the simulation of compressible flow past a cylinder (Re=100, M=0.5)
• Top: Mesh of full domain for simulation of compressible, Mach 0.5, Re=10,000 flow past a NACA 0012 airfoil at zero angle of attack to the mainstream flow, Middle: Mesh around body and wake, Bottom Left: Close up of the airfoil, Bottom Right: Close up of part of the wake region.
• The wake region (from $\frac{x}{L} = 2$ to $\frac{x}{L} = 5$) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re=10,000, 3307 triangles, 4180 quadrilaterals, N=11. Instantaneous iso-contours of the density are shown.
• The wake region (from $\frac{x}{L} = 2$ to $\frac{x}{L} = 5$) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re=10,000, 3307 triangles, 4180 quadrilaterals, N=11. Instantaneous iso-contours of the divergence of momentum are shown.
• The wake region (from $\frac{x}{L} = 2$ to $\frac{x}{L} = 5$) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re=10,000, 3307 triangles, 4180 quadrilaterals, N=11. Instantaneous iso-contours of the curl of momentum are shown.
• Hybrid Mesh and close ups for the simulation of compressible flow past a multi-body wing
• Iso-Mach contours and streamlines for M=0.5 flow past a two-dimensional multi-component wing.
• Skeleton mesh for flow past a three-dimensional NACA0012 airfoil with endplates.
• Iso-contours for x-component of momentum for M=0.5 flow past a three-dimensional NACA0012 airfoil with endplates.
• Magnetohydrostatic test case for the compressible code. Top Left: Mesh KTri = 38, KQuad = 22 Top Right: Magnetic streamlines of steady solution at N=12, Bottom: dependence of steady state error on expansion order.
• Three-dimensional magnetohydrostatic test case for the compressible code. Iso-contours of the magnetic and energy fields are shown at time t=1. Top Left: Tetrahedral mesh used, Top Right: x-component of the magnetic field, Middle Left: y-component of the magnetic field, Middle Right: z-component of the magnetic field, Bottom: Convergence plot showing exponential decrease in L2 error with increasing expansion order.
• Mesh used for the compressible Orszag-Tang vortex simulations on a structured mesh.
• Compressible Orszag-Tang Vortex (t=1, instantaneous fields, Mach=0.5). Top left: density, Top right: energy, Middle left: x-component of momentum, Middle right: y-component of momentum, Bottom left: x component of magnetic field, Bottom right: y component of magnetic field.
• Compressible Orszag-Tang Vortex (t=1, instantaneous fields, Mach=0.5). Top left: curl of momentum, Top right: divergence of momentum, Middle left: curl of magnetic field, Middle right: divergence of magnetic field, Bottom left: momentum streamlines, Bottom right: magnetic field streamlines.
• Compressible Orszag-Tang vortex triangle mesh with K=132.
• Compressible Orszag-Tang Vortex (t=1, instantaneous fields, Mach=0.2, K=132). Top Left: Curl of momentum along the diagonal(N=4), Botton Left: Iso-contours of curl of momentum(N=4), Top Right: Curl of momentum along the diagonal(N=6), Bottom Right: Iso-contours of curl of momentum(N=6)
• Compressible Orszag-Tang Vortex (t=1, instantaneous fields, Mach=0.2, K=132). Top Left: Curl of momentum along the diagonal(N=10), Botton Left: Iso-contours of curl of momentum(N=10), Top Right: Curl of momentum along the diagonal(N=16), Bottom Right: Iso-contours of curl of momentum(N=16)
• Instantaneous iso-contours of the simulation fields for flow past a cylinder LV=100,LB=100,A=0.1 with a magnetic field. From the top: (1) density, (2) pressure, (3) x component of the velocity field, (4) y component of the velocity field
• Instantaneous iso-contours of the simulation fields for flow past a cylinder LV=100,LB=100,A=0.1 with a magnetic field. Top: x component of the magnetic field, Bottom: y component of the magnetic field
• Vertex labelling for the standard quadrilateral region
• Mass matrix for the continuous basis for quadrilaterals (N=15).
• Vertex labelling for the standard triangular region
• Mass matrix for the continuous basis for triangles (N=15).
• Vertex labelling for three-dimensional standard regions
• Mass matrix for the continuous basis for hexahedra (N=15).
• Mass matrix for the continuous basis for prisms (N=15).
• Mass matrix for the continuous basis for tetrahedra (N=15).
• Vertex labelling for the (a) standard quadrilateral and (b) standard Hexahedral region