LA-UR-18-24126 Approved for public release; distribution is unlimited.
Title:
A comparative study of two different methods for RZ axisymmetric coordinates in context of Lagrangian discontinuous Galerkin hydrodynamics
Author(s):
Liu, Xiaodong Morgan, Nathaniel Ray Burton, Donald E.
Intended for:
AIAA Aviation and Aeronautics Forum and Exposition (AIAA AVIATION 2018), 2018-06-25/2018-06-29 (Atlanta, Georgia, United States)
Issued:
2018-05-11 (Draft)
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A comparative study of two different methods for RZ axisymmetric coordinates in context of Lagrangian discontinuous Galerkin hydrodynamics Xiaodong Liu∗, Nathaniel R. Morgan†, and Donald E. Burton‡ Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM, USA We present a comparative study of two different Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials while the density is advanced by a modal density evolution.23 Two different approaches are used to discretize the evolution equations ,i.e., the true volume approach and the areaweighted approach. For the true volume approach, the DG equations are derived using on the true 3D volume that is consistent with the geometry conservation law (GCL). A multidirectional approximate Riemann problem at the element surface nodes can be solved using the surface areas in axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based in the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, a multidirectional approximate Riemann problem identical to the one in 2D Cartesian geometry can be solved. This area-weighted approach conserves mass, and in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. A suite of test problems are calculated to demonstrate stable mesh motion and the expected second order accuracy of these methods.
Nomenclature J j w an V0 AN Ω Λs (x, y) (R, Z) (X, Y ) (ξ, η) u τ ρ σ
Jacobian matrix det(J) Volume of the current element Surface area normal of current element Volume of the initial element Surface area normal of initial element Reference element volume Surface area normal of reference element Natural coordinates Natural RZ axisymmetric coordinates Initial coordinates Reference coordinates Velocity Specific total energy Density Stress tensor
p Pressure S Source term e Specific internal energy t Time M Mass matrix ψ Vector of basis functions Superscript ∗ Riemann solution at an element node k A vector of Taylor expansion coefficients n Temporal value Subscript p An element node c An element corner i An element surface segement g A Gauss quadrature point
∗ Post-doc
research associate, X-Computational Physics Division. Corresponding author:
[email protected] X-Computational Physics Division. ‡ Scientist, X-Computational Physics Division. † Scientist,
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I.
Introduction
The modal Discontinuous Galerkin (DG)11, 12 hydrodynamic method is a promising alternative to the finite volume (FV) hydrodynamic methods. As in classical finite element methods, the accuracy for DG is obtained by using high-order polynomials within an element (also called a cell), rather than by using wide stencils as in the case of FV CCH methods. The DG(P) method with P > 0 can be regarded as the natural extension of FV methods to higher order methods. The modal DG method11, 12, 28 has been widely studied for high-order solutions of the gas dynamics equations in the Eulerian framework.6–8, 24–26, 45, 47 Likewise, DG hydrodynamic methods have been developed for Lagrangian motion in Cartesian coordinates.16, 19–21, 23, 34, 35, 42, 43 In addition, Liu and Morgan et. al.21 have developed the first true volume Lagrangian DG hydrodynamic method for axisymmetric coordinates. Following that, a comprehensive study of the true volume and area-weighted Lagrangian DG hydrodynamic method for axisymmetric coordinates has been reported in.22 Further research is explored on developing Lagrangian DG hydrodynamic methods for compressible flows in axisymmetric coordinates. In this paper, we present a comparative study of the area-weighted and true volume Lagrangian DG hydrodynamic method for axisymmetric coordinates. Different from,22 the modal density evolution is used to advance the density field. The velocity and specific total energy fields are approximated with Taylor series polynomials (modal DG method) about the center of mass of a reference element and the density is an expansion about the centroid of the reference element. The true volume DG hydrodynamic method conserves mass, momentum, and total energy; likewise, the method satisfies the GCL. However, this method cannot preserve spherical symmetry on an equal-angle polar grid for a 1D radial flow. The new area-weighted DG hydrodynamic method conserves mass and will preserve the spherical symmetry on an equal-angle polar grid for a 1D radial flow. This method, however, does not satisfy the GCL and will only conserve the physical momentum, and physical total energy in the limit of zero mesh size. An explicit second-order TVD RungeKutta method is employed for time marching. A limiting strategy is presented that yields accurate results and ensures monotonicity in the density, velocity, pressure, and total energy fields. The layout of the paper is as follows. The governing equations and nomenclature are described in Section II. The details on this Lagrangian DG methods for axisymmetric coordinates are presented in Section III. The discrete evolution equations for the specific volume, velocity, and specific total energy are presented in Section IV. The details of the multidirectional approximate Riemann solver and the analysis of the spherical symmetry preservation are discussed in Section V. The limiting of the nodal and modal fields is discussed in Section VI. The results from a large suite of test problems are reported in Section VII. Concluding remarks are given in Section VIII.
II.
Governing equations
The analytic Lagrangian equations (Fig. 1) for evolving specific volume, velocity, and specific total energy in axisymmetric coordinates are, dv RZ = ∇RZ · u RZ dt
(1)
d(u RZ )T = ∇RZ · σRZ dt
(2)
ρRZ ρRZ
dτ RZ = ∇RZ · (σ · u)RZ (3) dt The superscript RZ denotes that the respective variables are for 2D axisymmetric coordinates R = (Z, R). And v is the specific volume, ρ is the density, u is the velocity, τ is the specific total energy (τ = e + k), and σ is the stress tensor. The pressure, specific internal energy and specific kinetic energy are denoted as p, e, and k respectively. For gas dynamics, the stress is given by σ = −pI, where the pressure is calculated using an equation of state (EOS) for the material, p = EOS(ρ, e). The derivations in this work are valid for solid dynamics and the gas dynamics equations can be regarded as a special case. The time derivatives are total derivatives that move with the flow. The rate of change of the position is, ρRZ
dRRZ = u RZ dt 2 of 30 American Institute of Aeronautics and Astronautics
(4)
R φ Computational mesh
Z
Figure 1: Here, Z, R and ϕ denote the axial, radial and azimuthal direction in the cylindrical coordinate system respectively. The axisymmetric assumption means that there is no variation of the solution in the ϕ direction, so a 2D mesh can be rotated about the Raxis Z. A single element is shown that is rotated about the axis Z. The volume of this element is w = ϕ RdRdZ, where ϕ = 2π is a full rotation around the Z w
axis.
2D Cartesian-like velocity u RZ and stress σRZ are defined by, " # u RZ u ≡ Z , uR " # σZZ σZR RZ σ ≡ . σRZ σRR
(5)
(6)
Accordingly, the divergence operators on the 2D Cartesian-like velocity and stress tensor are ∇RZ · u RZ = " ∇
RZ
RZ
·σ
=
1 ∂ ∂ (uZ ) + (RuR ) ∂Z R ∂R
∂ 1 ∂ ∂Z (σZZ ) + R ∂R (RσZR ) σϕϕ ∂ 1 ∂ ∂Z (σRZ ) + R ∂R (RσRR ) − R
(7) #
T
(8)
∂ 1 ∂ (σZZ uZ + σZR uR ) + R(σRZ uZ + σRR uR ) (9) ∂Z R ∂R In order to describe the matrix manipulation in the paper more easily and clearly, the divergence ∇ is a row vector, and the velocity u is a column vector. In this work, we present DG methods for solving the axisymmetric equations given by Eqs. (1) - (3) using the true volume and the area-weighted approaches. A convention is followed throughout the paper for denoting scalars, vectors and tensors. A slanted character is used to denote a scalar such as density ρ, specific total energy τ , and specific internal energy e. A slanted bold character is used to denote a vector such as velocity u, surface normal vector (n or s), Taylor basis function ψ, and Riemann forces F ∗i . An upright character denotes a tensor such as stress σ, deformation gradient F, and Jacobian matrix J. At times it is useful to derive very general equations for an arbitrary unknown quantity that might be a scalar or a vector; in this case, a blackboard bold character U is used to denote the arbitrary unknown. The fluxes in a generalized evolution equation will be denoted as H, which can be either a vector or tensor. An arbitrary array of scalars or an array of vectors is denoted as Uk . Finally, we need to mention that in this work we follow the convention of matrix manipulation rather than the index notation. ∇RZ · (σ · u)RZ =
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III.
Discretization
The solution domain is decomposed into non-overlapping discrete control volumes called cells or elements. The initial element volume is w0 , where the superscript 0 denotes the initial time t = t0 . Each element has a constant mass, dm dt = 0. The elements will deform with the flow and the volume at a later time will be w(t) (Fig. 2). In the axisymmetric coordinates R and Z denote the radial and axial coordinates respectively. The element volume w(t) can be calculated by rotating the 2D area A (Fig. 2) around the axis Z. Since the problem is rotationally symmetric, the rotation angle about the Z axis cancels so it is dropped from the respective equations. Therefore, dw = RdA where dA = dRdZ. The coordinate system for the initial configuration of the element is (X, Y )T and the mapping from the initial configuration to the current configuration is given by R = R (X, t) . The deformation gradient is F = configuration is
∂R ∂X .
An isoparametric mapping from a reference element Ω to the current
R = R (ξ, t) =
X
bp R p .
p
Rp are the nodal coordinates of the element, bp are the canonical shape functions at the nodes (for the element type), and the reference coordinates are (ξ, η). The Jacobian matrix is J = ∂R ∂ξ . Both the deformation gradient and the Jacobian matrix (F and J respectively) will vary in time.
Figure 2: The map from the initial configuration and the map from a reference element are graphically illustrated.
A.
Taylor expansion
Unknown fields (e.g., v, u and τ ) can be represented with Taylor expansions on the reference element Ω about the center of mass. The Taylor expansion for an unknown U is ∂U ∂U U(ξ) = Ucm + (ξ − ξcm ) + (η − ηcm ) (10) ∂ξ cm ∂η cm The subscript cm denotes the center of mass. The Taylor basis functions, for a linear expansion about the center of mass, are h i ψ = 1 ξ − ξcm η − ηcm (11)
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A Taylor expansion of an unknown (with true volume and area-weighted approaches) can be succinctly written as, U(ξ, t) = ψ (ξ) · Uk (t)
(12)
where
Ucm Uk = ∂U ∂ξ |cm ∂U ∂η |cm
(13)
The Taylor expansion coefficients Uk are spatially constant on each element but they vary temporally. Uk is an array of scalars for the specific volume and specific total energy, τcm vcm ∂τ k (14) v k = ∂v ∂ξ |cm , τ = ∂ξ |cm ∂v ∂τ ∂η |cm ∂η |cm and Uk is an array of vectors for the velocity
vcm ∂v ∂ξ |cm
ucm ∂u k u = ∂ξ |cm ∂u ∂η |cm B.
(15)
∂v ∂η |cm
Center of mass
For the true volume approach, the center of mass is given by R ρξdw
w(t) ξcm = R
ρdw
and
R
w(t) ηcm = R
w(t)
ρηdw ρdw
;
(16)
w(t)
The basis functions in the vector ψ are constant in time and are solely a function of the reference coordinates, which are temporally constant, as such, dψ =0 dt The center of mass for the area-weighted approach is R ρξdA
ξcm
A(t)
= R
ρdA
and
A(t)
ηcm
(17)
R
ρηdA
A(t)
= R
ρdA
.
(18)
A(t)
The planar center of mass with the area weighted approach changes temporally, so the basis functions will also change temporally dψ dt 6= 0. Additional terms are included in the area-weighted DG evolution equations to account for the temporal variation in the basis functions. C.
Nodal fields
The previous subsection addressed the modal fields that are approximated with linear Taylor expansions. The density, specific internal energy, and pressure fields are represented as nodal fields. These fields are calculated at the Gauss quadrature points Rg and at the corners of the element Rc = Rp . The density at a particular location ξ j inside the reference element is calculated by taking the inverse of the specific volume 2 ρj = 1/v(ξ j , t). The specific internal energy at a particular location is ej = τ (ξ j , t) − 1/2 u(ξ j , t) . The pressure at a particular location is pj = EOS(ρj , ej ). The pressure and velocity values in the element corner are used in the multidirectional approximate Riemann solver (Section V).
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D.
A true volume discontinuous Galerkin method
In this section, we present a DG discretization of the governing evolution equations using the true volume approach. Based on Eq. (7), (8) and (9), we introduce specific expressions for the true volume approach as follows, ∇T V · u T V =
∇ T V · σT V
1 ∂ 1 ∂ ∂ ∂ (uZ ) + (RuR ) = ( (RuZ ) + (RuR )) ∂Z R ∂R R ∂Z ∂R " " # T # T 1 ∂ ∂ ( (Rσ ) + (Rσ )) 0 ZZ ZR R ∂Z ∂R = − σϕϕ 1 ∂ 1 ∂ R ( ∂Z (RσRZ ) + R ∂R (RσRR )) R
(19)
(20)
1 ∂ 1 ∂ R(σZZ uZ + σZR uR ) + R(σRZ uZ + σRR uR ) (21) R ∂Z R ∂R where the superscript T V denotes the true volume approach. The analytic Lagrangian equations for evolving specific volume, velocity, and specific total energy in 2D axisymmetric coordinates are, ∇T V · (σ · u)T V =
dv T V = ∇T V · u T V dt
(22)
d(u T V )T = ∇T V · σT V dt
(23)
ρT V ρT V
dτ T V = ∇T V · (σ · u)T V (24) dt Here, the superscript T V will be dropped from the derivation that follows. The governing evolution equations for evolving v, u and τ have the generalized form, ρT V
dU =∇·H dt where U is the respective unknown field (scalar or a vector) and H is the flux (vector or a tensor). The evolution equation is multiplied by the Taylor basis functions ψ and then integrated over the current element configuration. Z dU ψT ρ − ∇ · H dw = 0 dt ρ
w(t)
which creates a system of q equations for q unknown Taylor expansion coefficients. The volume of the element is dw = RdRdZ for axisymmetric coordinates. Substituting a Taylor expansion into the above equation gives, Z Z dUk ρψ T ψdw · = ψ T (∇ · H)dw (25) dt w(t)
w(t)
For the term on the left side of Eq. (25), the basis functions were factored out of the time derivative because dψ/dt = 0; likewise, the time derivative of the basis coefficients dUk /dt was factored out of the integral because they are solely a function of time for each element. The integral on the left side of Eq. (25) is the mass matrix, Z Z T M= ρψ ψdw = ρψ T ψRjdΩ (26) Ω
w(t)
The integral in the mass matrix is transformed to the reference element Ω because the basis expansions are defined on the reference element. Using integration by parts with the term on the right side of Eq. (25), the evolution of the unknown basis coefficients is,
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dUk M· = dt
Z
Z
T
∇ · (ψ H)dw − w(t)
(∇ψ T ) · Hdw
w(t)
For a linear Taylor expansion, ∇ · (ψ1 H) ∇ · (ψ T H) = ∇ · (ψ2 H) ∇ · (ψ3 H)
∇ψ1 ∇ψ T = ∇ψ2 . ∇ψ3
and
Transforming the volume integral from the current element to the reference element, Z Z dUk T M· ∇ξ ψ T · jJ−1 · HRdΩ ∇ · (ψ H)dw − = dt
(27)
Ω
w(t)
The first term of the right hand side of Eq. (27) will be converted to a surface integral using Gauss’s theorem that relates the integral of the divergence over a volume w(t) with the flux through the surface ∂w(t). For cylindrical coordinates, Gauss’s theorem for a vector in axisymmetric coordinates is Z I TV TV ∇ · h RdRdZ = n · h T V Rda (28) A(t)
∂A(t)
and for a tensor is Z
I
∇T V · σT V RdRdZ =
A(t)
n · σT V Rda −
∂A(t)
Z σϕϕ e R dA
(29)
A(t)
Here, e R = [0 1], n is a row vector denoting the surface normal vector in the natural coordinates, and da is a differential segment on the boundary of A(t). Using Gauss’s Theorem for axisymmetric coordinates (Eqs. 28 and 29), the resulting evolution equations for the unknown basis coefficients for specific volume v k , velocity u k and specific total energy τ k are, Z I dv k M· ψ T (n · u ∗ )Rda − ∇ξ ψ T · jJ−1 · uRdΩ = (30) dt Ω
∂A(t)
du k M· = dt
I
T
Z ψ e R σϕϕ dA − ∇ξ ψ T · jJ−1 · σRdΩ T
ψ (n · σ )Rda − ∂A(t) k
M·
∗
Z
dτ = dt
A(t)
I
∗
ψ T n · (σ∗ · u )Rda −
(31)
Ω
Z
∇ξ ψ T · jJ−1 · (σ · u)RdΩ
(32)
Ω
∂A(t)
The first term on the right hand side in Eq. (30), (31), and (32) requires solving a Riemann problem on the surface of the deformed element ∂A(t). The Riemann velocity and stress are denoted with a superscript ∗. The Lagrangian DG evolution equations in Eqs. (30) - (32) are very general and no assumptions were made concerning the Riemann solver, the approximation of the integrals, or the order of the Taylor expansion. The Riemann solver in this work is at the surface nodes of the element and follows the Lagrangian finite volume methods (Section V). The Lagrangian DG method reduces to a first-order finite volume method when using only the element average in the Taylor expansion.
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E.
An area-weighted discontinuous Galerkin method
In this section, we present a DG discretization of the governing evolution equations using the area-weighted approach. Based on Eq. (7), (8) and (9), we introduce specific expressions for the area-weighted approach as follows, ∇AW · u AW = ∇2D · u 2D + " ∇
AW
·σ
AW
=∇
2D
·σ
2D
+
uR R
σZR R σRR −σϕϕ R
∇AW · (σ · u)AW = ∇2D · (σ2D · u 2D ) +
(33) #
T
(34)
σRZ uZ + σRR uR R
(35)
where, ∂ ∂ (uZ ) + (uR ) ∂Z ∂R " # ∂ ∂ (σZZ ) + ∂R (σZR ) ∂Z = ∂ ∂ ∂Z (σRZ ) + ∂R (σRR )
∇2D · u 2D =
∇
2D
2D
·σ
T
∂ ∂ (σZZ uZ + σZR uR ) + (σRZ uZ + σRR uR ) ∂Z ∂R The superscript AW denotes the area-weighted approach. The analytic Lagrangian equations for evolving specific volume, velocity, and specific total energy in axisymmetric coordinates are, ∇2D · (σ2D · u 2D ) =
dv AW = ∇AW · u AW dt
(36)
d(u AW )T = ∇AW · σAW dt
(37)
ρAW ρAW
dτ AW = ∇AW · (σ · u)AW (38) dt Here, the superscript AW will be dropped from the derivation that follows. The governing evolution equations for evolving v, u and τ have the generalized form, ρAW
ρ
SU dU = ∇2D · H2D + dt R
(39)
where U is the respective unknown field (scalar or a vector) and H is the flux (vector or a tensor). From Eqs. (33), (34) and (35), the source terms are h i Sν = uR , Su = σZR σRR − σϕϕ , Sτ = σRZ uZ + σRR uR . (40) The evolution equation is multiplied by the Taylor basis functions ψ and then integrated over the current element configuration. Z dU SU T 2D 2D ψ ρ −∇ ·H − dw = 0 dt R w(t)
which creates a system of q equations for q unknown Taylor expansion coefficients. The volume of the element is dw = RdA = RdRdZ for axisymmetric coordinates. Substituting a Taylor expansion into the above equation gives, Z Z Z SU dU ψ T (∇2D · H2D )dw + ψT dw ψ T ρ dw = dt R w(t)
w(t)
w(t)
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¯ By introducing an element average radius R, R
dw
¯ = w(t) R R
(41)
dA
A(t)
we can arrive at Z
¯ R
Z
dU ¯ ψ ρ dRdZ = R dt T
A(t)
T
ψ (∇
2D
·H
2D
Z )dRdZ +
A(t)
ψ T SU dRdZ
(42)
A(t)
Because dψ/dt 6= 0, thus, an extra term is present on the right hand side to account for the rate of change of the basis functions.
¯ R
Z
ρψ T ψdRdZ ·
dUk ¯ =R dt
Z
ψ T (∇2D · H2D )dRdZ +
¯ ψ T SU dRdZ − R
Z
ρψ T
dψ dRdZ · Uk (43) dt
A(t)
A(t)
A(t)
A(t)
Z
The integral on the left side of Eq. (43) is the area-weighted mass matrix, Z Z T AW ¯ ¯ ρψ ψdA = R ρψ T ψjdΩ M =R
(44)
Ω
A(t)
The integral in the area-weighted mass matrix is transformed to the reference element Ω because the basis expansions are defined on the reference element. Using integration by parts with first term of the right hand side of Eq. (43), the evolution of the unknown basis coefficients is,
MAW ·
dUk ¯ =R dt
Z
Z
¯ ∇2D ·(ψ T H2D )dRdZ−R
A(t)
(∇2D ψ T )·H2D dRdZ+
A(t)
Z
¯ ψ T SU dRdZ−R
A(t)
Z
ρψ T
dψ dRdZ·Uk dt
A(t)
For a linear Taylor expansion, ∇2D · (ψ1 H2D ) ∇2D · (ψ T H2D ) = ∇2D · (ψ2 H2D ) ∇2D · (ψ3 H2D )
and
∇2D ψ1 ∇2D ψ T = ∇2D ψ2 . ∇2D ψ3
Using Gauss’s Theorem for 2D Cartesian coordinates, the resulting evolution equations for the unknown basis coefficients for specific volume v k , velocity u k and specific total energy τ k are,
MAW ·
dv k ¯ =R dt
I
¯ ψ T (n · u ∗ )da − R
du k ¯ =R dt
I
¯ ψ T (n · σ∗ )da − R
M
dτ k ¯ · =R dt
I
∗
∗
¯ −R
ρψ T
Ω
Z Z ¯ ρψ T dψ jdΩ · u k (46) ∇ξ ψ T · jJ−1 · σdΩ + ψ T Su jdΩ − R dt
¯ ψ n · (σ · u )da − R T
Ω
Z
∇ξ ψ
T
· jJ
−1
Ω
Z · (σ · u)dΩ +
ψ T (σRZ uZ + σRR uR )jdΩ
Ω
Ω
∂A(t)
Z
Z
Ω
Ω
∂A(t)
AW
Z Z ¯ ρψ T dψ jdΩ · v k (45) ∇ξ ψ T · jJ−1 · udΩ + ψ T uR jdΩ − R dt
Ω
∂A(t)
MAW ·
Z
dψ jdΩ · τ k dt
Ω
(47) 9 of 30 American Institute of Aeronautics and Astronautics
h where Su = σZR
i σRR − σϕϕ . Similar to the DG discretization with the true volume approach, the RieR ¯ ρψ T dψ jdΩ· mann problem is solved at the surface nodes of the element (Section V). For the source terms, R Ω
¯ v , R k
R Ω
ρψ T dψ dt jdΩ
¯ · u , and R k
R Ω
ρψ T dψ dt jdΩ
k
· τ , the total derivative
dψ dt
dt
is calculated by finite difference
method numerically.44 Eqs. (45), (46), and (47) have the same form as those in 2D Cartesian coordinates23 except for the last two source terms of the right hand side. F.
Mass conservation
For the true volume approach, mass conservation is enforced by guaranteeing that the mass matrix (Eq. 26) is constant in time. The total derivative of the mass matrix is R R d T dM d ρψ T ψRjdΩ = dt ρψ ψRj dΩ dt = dt Ω (48) RΩ T d d = (ψ T ψ) dΩ ψ ψ dt (ρRj) + ρRj dt Ω
The terms inside the integral are dψ T ψ =0 dt d (ρRj) =0 dt
Substituting these relations into Eq. (48) gives, dM =0 dt
(49)
For the area-weighted approach, the area-weighted mass matrix MAW is not constant in time because the basis functions change temporally. The area-weighted mass matrix is updated every time integration step. The mass of the element is conserved because. Z Z 0 0 ¯ ¯ R(t ) ρ(t )dRdZ = R(t) ρ(t)dRdZ = m (50) A(t0 )
A(t)
The density expansion is about the centroid of the planar area A(t). Usually, when the specific volume is advanced for the density field,22 by guaranteeing the Eq. 49 or 50, the mass conservation can be preserved for the the true volume or area-weighted approach. Several ways have been explored to guarantee the mass conservation in.22 A constant mass matrix requires that Z Z 0 0 0 ρ(ξ, t )ψψR(ξ, t )j(ξ, t )dΩ = ρ(ξ, t)ψψR(ξ, t)j(ξ, t)dΩ Ω
Ω
Mass conservation can be enforced at every point in space by requiring ρ(ξ, t0 )j(ξ, t0 ) = ρ(ξ, t)j(ξ, t)
(51)
which is the strong mass principle.15, 23 Mass conservation can also be enforced in a variational manner, such as guaranteeing that the integral of a modal distribution of density over the element is constant (Section A). The subscript z denotes the centroid. G.
Temporal integration
A second-order explicit Runge-Kutta (RK) method is used to evolve the system of equations from time level n to n + 1. The time integration, for a vector of unknown Taylor expansion coefficients, is
Uk
n+1
s1 n n Uk = Uk + ∆t M−1 · Rn n s1 1 s1 s1 = 21 Uk + 21 Uk + 2 ∆t M−1 ·R
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(52)
The superscript s1 denotes the solution after the first stage. For the true volume approach, The mass matrix n s1 M is constant in time (Section F) so M−1 = M−1 . The position of a surface node x p is updated using, s1
(Rp )
n+1
n
(Rp ) = (Rp ) + ∆t(u ∗p )n n s1 = 12 (Rp ) + 21 (Rp ) + 12 ∆t(u ∗p )s1
(53)
u ∗p (Section V) denotes the Riemann velocity at a surface node.
IV.
Evolution of physical fields
For the area-weighted approach, the discretization Eq. (45), (46), and (47) are almost identical to the Lagrangian DG evolution equations for 2D Cartesian coordinates23 except for the last Rtwo source terms. For example, the specific volume evolution equation has two source terms that are ψ T uR jdΩ Ω R R ¯ ρψ T dψ jdΩ · v k , it can be treated using the ¯ ρψ T dψ jdΩ · v k respectively. For the source term R and R dt dt Ω Ω R ¯ same way as the volume integral R ∇ξ ψ T · jJ−1 · udΩ. However, in order to preserve the spherical Ω R symmetry on an equal-angle grid, we have to deal with the source term ψ T uR jdΩ carefully. In Section B, Ω
we will discuss how to calculate the this source term to guarantee symmetry preservation. Therefore, the subsections that follow will focus on the evolution of the physical fields with the true volume approach. A.
True volume density evolution
The DG method in this work evolves a Taylor expansion of the density field ρ over the reference element Ω. ˆ (ξ) · ρk (t) ρ (ξ, t) = ψ The Taylor basis functions for the density field are about the element centroid, h i ˆ = 1 ξ−ξ η−η ψ z z
(54)
(55)
The modal density approach evolves a Taylor expansion (Eq. 54) of the density field that guarantees mass conservation. Z Z ˆ T ρ(ξ, t0 )R0 j 0 dΩ = ψ ˆ T ρ(ξ, t)RjdΩ ψ (56) Ω
Ω
The determinant of the Jacobian matrix for the initial configuration is denoted as j 0 . Substituting the initial and current density Taylor expansions into Eq. (56) gives, Z Z T ˆ ψRjdΩ ˆ ˆ T ρ(ξ, t0 )R0 j 0 dΩ ψ · ρk (t) = ψ Ω
Ω
The Taylor expansion coefficients for the density field ρk at the current time are, −1
ρk (t) =
Z
ˆ T ψRjdΩ ˆ ψ
Z ·
Ω
B.
ˆ T ρ(ξ, t0 )R0 j 0 dΩ ψ
(57)
Ω
True volume momentum evolution
The DG method in this work evolves a Taylor expansion of the velocity field u over the reference element Ω. u (ξ, t) = ψ (ξ) · u k (t)
(58)
Here, u k is an array of vectors. Using the DG discretization in Eq. (31), the system of equations for evolving the velocity field is, 11 of 30 American Institute of Aeronautics and Astronautics
0 0 P P u cm (ρψ2 ψ2 )g Rg jg Ωg (ρψ2 ψ3 )g Rg jg Ωg ∆ ∂u · ∆t ∂ξ |cm = g∈Ω g∈Ω P P ∂u (ρψ3 ψ2 )g Rg jg Ωg (ρψ3 ψ3 )g Rg jg Ωg ∂η |cm g∈Ω g∈Ω P P ai Ri n i · σ∗c (ψ1 jσϕϕ e R )Ωg 0 g∈Ω i∈w(t) P (∇ ψ · jJ−1 · σR) Ω P P ξ 2 g g ψ2i ai Ri n i · σ∗c (ψ2 jσϕϕ e R )Ωg = − g∈Ω − g∈Ω i∈w(t) P −1 P P (∇ξ ψ3 · jJ · σR)g Ωg (ψ jσ e )Ω ψ a R n · σ∗ ρ¯w 0 0
3i i
i
i
i∈w(t)
c
3
ϕϕ R
g
(59)
g∈Ω
g∈Ω
where 0 is a row vector [0 0]. ρ¯ denotes the element average density and w is the current element volume R RdRdZ. The subscript g denotes Gauss point locations in the reference element, subscript p denotes the A
value at an element node, and subscript i denotes the element surface segments connected to a node. ai Ri n i denotes the surface area normal vector of a segment i. The nomenclatures for the surface segments connected to a node p is shown in Fig. 3. The contribution of node p to the second term of right hand side is expressed as, 2 P P ∗ ai Ri n i · σ∗c a R n · σ i i i c i=1 i∈w(t) P 2 P ∗ ∗ ψ2i ai Ri n i · σc (60) ψ a R n · σ = 2i i i i c . i∈w(t) i=1 P 2 P ψ3i ai Ri n i · σ∗c ψ3i ai Ri n i · σ∗c i∈w(t) p
Here, a1 =
1 2 ap−p ,
a2 =
1 + 2 app ,
i=1
and
3Rp + Rp− Rp + Rp− ψ 1p + ψ1p− , 6 6 Rp + Rp− 3Rp + Rp− ψ 2p + ψ2p− , = 6 6 3Rp + Rp− Rp + Rp− = ψ 3p + ψ3p− , 6 6
3Rp + Rp+ Rp + Rp+ ψ 1p + ψ 1p + , 6 6 3Rp + Rp+ Rp + Rp+ = ψ 2p + ψ 2p + , 6 6 3Rp + Rp+ Rp + Rp+ = ψ 3p + ψ 3p + . 6 6
R1 ψ11 =
R 2 ψ 12 =
R1 ψ21
R 2 ψ 22
R1 ψ31
R 2 ψ 32
Here, ap−p and app+ denote the length for the face p− p and pp+ respectively. For the DG(P1) method, it is known that R and ψ vary linearly along any face of an element. Assuming that Riemann stress varies linearly along any face of an element, we could get the above parameters using analytical integration of the first term on the right hand side of Eq. (31). These parameters are also used in the total energy evolution (Section C). Such parameters make the flux discretization of our scheme compatible with the GCL.3, 32, 39, 46 The Riemann stress at an element node is σ∗c (Section V). Please refer to Section A for the description of the relevant variables. C.
True volume total energy evolution
The DG method in this work evolves a Taylor expansion of the total energy field τ over the reference element Ω. τ (ξ, t) = ψ (ξ) · τ k (t)
(61)
Using the DG discretization in Eq. (32), the system of the equations for the evolution of the total energy is,
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Figure 3: A patch of four elements is shown in the natural coordinate system. The element surface area is decomposed into smaller segments, where the surface area normal vector of a segment is ai Ri n i in the natural coordinates. A multi-direction approximate Riemann problem is solved at the surface nodes of the element. The control surface for the Riemann solver is constructed from all segments connected to the node p and is denoted as i ∈ p. The control surface for the Riemann solver is highlighted in the figure with a dashed-line. The inputs to the Riemann solver (e.g., velocity and stress) are the element corner values and they are constant over an element corner. The corner values are a function of the Taylor expansion of the unknowns evaluated at the node p. These expansions are about either the element center of mass cm or the centroid z.
ρ¯w 0 0
0 τcm (ρψ2 ψ3 )g Rg jg Ωg ∆ ∂τ · ∆t ∂ξ |cm = g∈Ω g∈Ω P P ∂τ (ρψ3 ψ2 )g Rg jg Ωg (ρψ3 ψ3 )g Rg jg Ωg ∂η |cm g∈Ω g∈Ω P ai Ri n i · σ∗c · u ∗p 0 i∈w(t) P (∇ ψ · jJ−1 · σ · uR) Ω P ∗ ∗ ξ 2 g g · u ψ a R n · σ 2 i i i p c i = − g∈Ω P i∈w(t) −1 P (∇ ψ · jJ · σ · uR) Ω ∗ ∗ ξ 3 g g ψ3i ai Ri n i · σc · u p g∈Ω P
0 (ρψ2 ψ2 )g Rg jg Ωg
P
(62)
i∈w(t)
The Riemann velocity at an element node is u ∗p (Section V). Refer to Section B for the relevant variables.
V.
Riemann solver
Several effective multidirectional approximate Riemann solvers for Lagrangian finite volume CCH have been proposed2, 14, 29, 31 for gas (solid) dynamics problems. In this work, a multidirectional approximate Riemann problem is solved at the surface nodes of the element following the finite volume method in.2 The discussion that follows will first focus on the details of the multidirectional approximate Riemann solver in axisymmetric coordinates with the true volume approach followed by a discussion of the area-weighted approach. A.
True volume Riemann solver
Following the method in,35 the Riemann force acting on a segment of the element surface is, F ∗i = ai Ri n i · σ∗c = ai Ri n i · σc + µc ai Ri u ∗p − u c
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(63)
The nomenclature is illustrated in Fig. 3. The acoustic impedance is µc = ρc where c is the sound speed of the element. Momentum conservation at the surface node requires that the summation of all forces around the node to be equal to zero. X ∗ Fi = 0 (64) i∈p
A single Riemann velocity is found at the surface node that guarantees momentum conservation. P (µc ai Ri u c − ai Ri n i · σc ) i∈p P u ∗p = µc ai Ri
(65)
c∈p
Total energy is also conserved because, X
F ∗i · u p = ∗
i∈p
X
F ∗i · u ∗p = 0
(66)
i∈p
Analysis is presented in23 to show that the new Lagrangian DG method, combined with this Riemann solver, satisfies the second-law of thermodynamics. For the case of a 1D radial flow on an equal-angle polar grid that is shown in Fig. 4, it is easy to conclude that this true volume Riemann solver can not preserve the spherical symmetry because the nodal Riemann velocity u ∗p is not in the radial direction. B.
Area-weighted Riemann solver
¯ (Eq. 41). Dividing the area-weighted The area-weighted equations have a common element average radius R DG evolution equations (Eq. 45 - 47) by the element average radius produces the 2D Cartesian DG evolution equations with source terms. After this division, the surface fluxes do not have a radius so the multidirectional approximate Riemann solver with the area-weighted approach is identical to the one for 2D Cartesian coordinates.23 The area-weighted Riemann force and Riemann velocity are F ∗AW = ai n i · σ∗c = ai n i · σc + µc ai u ∗p − u c i P u ∗AW = p
i∈p
(67)
(µc ai u c − ai n i · σc ) P µc ai
(68)
c∈p
P ∗AW = The Riemann velocity was calculated by enforcing conservation in the area-weighted Riemann force Fi i∈p P ∗AW · u ∗AW = 0. 0, which in turn, guarantees the conservation of an area-weighted total energy flux Fi p i∈p
The area-weighted Riemann force is not equal to the physical force because it is missing a surface radius. However, in the limit of an infinitesimal mesh size, the surface radius is equal to the element average radius. As a result, the physical momentum and physical total energy are conserved in the limit that the mesh size goes to zero. For the case of a 1D radial flow on an equal-angle polar grid that is shown in Fig. 4, we will prove that this area-weighted Riemann solver can preserve the spherical symmetry. The following discussion will first consider the Riemann solver with the area-weighted approach followed by an analysis of the area-weighted DG evolution equations. The Riemann solver at the surface nodes is identical to the one used in 2D Coordinates. For a 1D radial flow on an equal-angle polar mesh, the theta direction of the Riemann velocity is u ∗pθ = 0, so the grid point moves radially. Then we proceed to analyze the area-weighted DG equations to show that it preserves spherical symmetry for this test case. For simplicity, we consider the case of gas dynamics so Su = 0. It is easy to conclude that the area-weighted DG momentum equation is symmetry preserving on an equal-angle polar mesh for a 1D radial flow. As for the area-weighted DG specific volume equation preserves spherical symmetry for this test case, from Eq. (45),
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Z Ω
ρψ T ψjdΩ ·
dv k = dt
I
ψ T (n · u ∗ )da −
∂A(t)
Z
Z Z dψ 1 ψ T uR jdΩ − ρψ T ∇ξ ψ T · jJ−1 · udΩ + jdΩ · v k ¯ dt R Ω
Ω
Ω
There are 2D Cartesian-like terms plus two source terms,
1 ¯ R
R
ψ T uR jdΩ and
Ω
R Ω
k ρψ T dψ dt jdΩ · v respectfully.
The 2D Cartesian R terms and the last source term guarantee symmetry preservation for this test case. The source term R1¯ ψ T uR jdΩ is discretized by using a single element average radial velocity u ¯R . Symmetry Ω √ preservation can be proven by demonstrating that this source term only depends on the radius r = R2 + Z 2 . R As an example, the source term R1¯ ψ T uR jdΩ for the element w3 in Fig. 4b is Ω Z Z Z |¯ u |(¯ r3z ) 1 1 T ψ T uR jdΩ = ψ T jdΩ (69) ψ uR jdΩ = ¯ 3z r¯3z sinθ3z r¯3z R Ω
Ω
Ω
p 2 + Z 2 and where the amplitude of the element average velocity |¯ u | only depends on the radius r¯3z = R3z 3z does not vary in the angular direction. Test problem results will be presented in Section VII to verify the spherical symmetry preservation of the area-weighted DG method with 1D radial flows on an equal-angle polar mesh. Future research could consider finding a more accurate treatment of this source term for a third-order or higher Lagrangian DG method.
(a) The nomenclatures associated with one point p(b) The nomenclatures associated with one element w3
Figure 4: The nomenclatures for 1D spherical flow on an equal-angle polar grid. The left one shows the nomenclatures associated with one point p. w1 , w2 , w3 and w4 are four elements surrounding the point p. pl , pb , pr and pt are four points surrounding the point p. The local orthonormal basis (e r , e θ ) at point p is used for simplifying the computation. Since the flow is symmetric in the angular direction, the density, pressure and velocity magnitude in the elements w1 and w4 are ρ1 , p1 , U1 , and in the elements w2 and w3 are ρ2 , p2 , U2 . In this section, the vector n ij denotes the unit normal vector for the face between element i and j and it δθ is directed from element i to j. In the local orthonormal basis (e r , e θ ), n 12 = (cos δθ 2 , −sin 2 ), n 23 = (0, 1), δθ n 34 = (−cos δθ 2 , −sin 2 ) and n 41 = (0, −1). The element average velocities of these four elements could be expressed as follows, U 1 = U1 n 12 , U 2 = U2 n 12 , U 3 = −U2 n 34 and U 4 = −U1 n 34 . The right one shows the nomenclatures associated with one element w3 . p3z is the volume average center q of the element w3 . Its ¯ ¯ ¯2 . coordinate is (Z3z , R3z ). The spherical radius of point p3z is r¯3z , defined by r¯3z = Z¯ 2 + R 3z
VI.
3z
Limiting
Limiting certain fields toward the element average value (i.e., piecewise constant over the element) is essential for monotone solutions on problems with shocks and discontinuities. A set of design goals will 15 of 30 American Institute of Aeronautics and Astronautics
be followed for creating a limiting approach for these Lagrangian DG methods. The first goal is to ensure that the pressure and velocity in an element corner is bounded by the neighboring element average values around the node, which ensures that the inputs to the Riemann solver are bounded. The second goal is to ensure that the evolved Taylor expansions (specific volume, velocity, and specific total energy) are within the permissible bounds defined by the neighboring elements around each node. The third goal is to guarantee second-order accuracy on smooth flows. A limiting approach will be described that satisfies these design goals. In this work, the velocity, specific volume, pressure and specific total energy fields are limited towards a constant field. The limiting is done after the pressure is calculated; in other words, the pressure is obtained using unlimited fields. The limiting on the specific total energy is only used in the temporal integration. All the limiting is done after each time integration step (Eq. 52), and the limited fields are used in the RK time integration. This limiting strategy has been verified and shown to achieve the above design goals.23 Some fields are only known at the discrete points inside the element (e.g., the pressure), while other fields are represented by a Taylor expansion (modal basis expansion). A limiting approach23 is presented that works for modal and nodal fields. Likewise, limiters are presented for scalar and vector fields that preserve spherical symmetry on equal-angle polar meshes for 1D radial flows with the area-weighted DG method and with the DG method in 2D Cartesian coordinates. A.
Scalar limiter, φ
The limiting of a scalar field in an element is achieved by, U (ξ)
lim
= U + φ U (ξ) − U
(70)
The superscript lim denotes the limited field, U is the element average value for the field, and U (ξ) is the unlimited field. The scalar limiter reduces the field to be equal to the element average with φ = 0; likewise, the field is unlimited when φ = 1. The scalar limiter is found using the Barth-Jesperson limiter,1 max η(U −U) min 1, U ξ −U if U ξ p − U > 0 ) ( p min φ= (71) −U) η(U min 1, U ξ −U if U ξ p − U < 0 ) ( p 1 else max
min
The subscript p denotes the location of a reference element node. The values, U and U , are the max and min element average values surrounding a node. The variable η is the maximum permitted difference and it is in the range of 0 to 1. A value of 0 corresponds to a first-order discretization and a value of 1 permits the largest variation of the field over the element. In,23 the effect of this adjustable coefficient η was investigated. A value of η ranging from 0.6 to 1.0 can guarantee the designed second-order on a smooth flow. A unique limiting coefficient is found for each variable. The smallest limiting coefficient is used for the entire element for the respective scalar field. B.
Tensor limiter for vector fields, φ
The limiting of a vector field in an element is achieved by, lim
U (ξ)
= U + φ · U (ξ) − U
(72)
φ is a limiting tensor and the superscript lim denotes the limited vector field. The limiting tensor φ smoothly transitions the vector field from a higher-order field (φ = I, the identity matrix) to the element average value (φ = 0, a matrix or zeros). The tensor limiter values are found by directional limiting, " # θ1 0 T φ=R · ·R (73) 0 θ2 The concept is to limit the field in a particular direction (e.g., the principle strain direction30 or the local flow direction33 ) and then to use a rotation matrix R to build the limiting tensor. The limiting coefficients θ1 and θ2 are calculated using the scalar Barth-Jesperson limiter shown in Eq. (71). The max and min neighboring 16 of 30 American Institute of Aeronautics and Astronautics
values around a node are calculated by rotating into the same limiting direction. The smallest calculated limiter coefficients are used. The merits of this approach is that it can preserve spherical symmetry and that higher-order solutions are possible in the tangential direction to a shock. In this paper, if not specified, limiting the field in the local flow direction is used. In addition, the limiting on the stress tensor in the principle strain direction has been investigated well.4
VII.
Test problems
In this section, a suite of test problems have been calculated to demonstrate the accuracy and robustness of the Lagrangian DG hydrodynamic methods. Smooth flow problems with analytical solution are used to quantify the convergence order of accuracy of the Lagrangian DG methods using the L2 error norm. Taking pressure p as an example, the L2 error norm is defined by, v sZ uN um Z uX 2 (p − pa ) dw = t (p − pa )2 dwi (74) ||p − pa ||L2 = w
i=1
wi
where pa denotes the analytical pressure, and N um is the number of elements in the problem. All test problems in this work use a gamma-law gas EOS, and unless noted otherwise, are calculated using a linear Taylor series polynomial that is denoted as DG(P1). In this work, various types of meshes are used to calculate a diverse suite of test problems. For a polar grid, 40 × 20 denotes the √ number of elements along the spherical radial (r) and angular (θ) directions. Here, r is defined by r = Z 2 + R2 , and θ denotes the tangential direction to the spherical radial direction. In addition to a full polar mesh defined with (r, θ) ∈ [rmin , rmax ] × [0, π2 ], some calculations will also be performed using a polar grid with (r, θ) ∈ [rmin , rmax ] × [ π4 , π2 ] that is termed a partial polar mesh. For the polar mesh, symmetric boundary conditions are imposed for the boundaries at θ = π4 (or θ = 0) and θ = π2 . In this work, for the true volume method (or the area-weighted method) on a polar mesh, TV (or AW) denotes the results on a partial polar mesh while TV(f) (or AW(f)) denotes the results on a full polar mesh. A.
Free expansion problem
The accuracy of the DG Lagrangian methods is tested by calculating the free expansion problem.10, 38 The material in this test problem involves an ideal gas with γ = 5/3. The initial conditions of this test problem are given as follows. ρ0 (r) = 1, u 0 (r) = 0, p0 (r) = 1 − r2 . This problem solution, √ has the following analytical 2 2t u(r, t) = 1+2t ρ(r, t) = r31(t) , p(r, t) = r51(t) (1 − r2r(t) ), ro (t) = 1 + 2t2 , 2 r, o
o
o
where ro is the radius of the free outer boundary and u(r, t) is the velocity vector in the spherical radial direction. A partial polar mesh is used for the calculations using the true volume and area-weighted DG methods. The initial grid and the one at t = 1 are shown in Figure 5. For this case, K and L represent the number of elements along the spherical radial and angular directions. In order to exclude the effect of 3 7 the free-surface boundary, the elements located in the interval [ 10 K + 1, 10 K] × [ 15 L + 1, 45 L] are used for 10 the error calculation following Cheng and Shu. The numerical error and convergence order of accuracy at t = 0.4 are shown in Table 1 and Table 2 using DG(P1). Both DG methods deliver the expected second-order accuracy. B.
Kidder shell
Kidder derived a series of analytical solutions for the isentropic compression of a gas.17, 32 In this section, we calculate an isentropic compression of a hollow shell of gas. A time-varying boundary condition on the inner and outer surfaces causes the shell to compress. The initial computational domain is defined by (r, θ) ∈ [0.9, 1] × [ π4 , π2 ] in polar coordinates for the calculations with the true volume and area-weighted DG methods (i.e., a partial polar mesh is used). The initial inner and outer radius are denoted by ri0 and ro0 respectively, where the subscripts i and o denote the inner and outer surface, and the superscript 0 denotes
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1
0.9 1.5 0.8
0.7
1.2
0.6
0.9 R 0.5
R
0.4 0.6 0.3
0.2 0.3 0.1
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.3
Z
(a) The mesh at t = 0
0.6 Z
0.9
(b) The mesh at t = 1
Figure 5: The meshes are shown for the free expansion problem using DG(P1). The meshes with the true volume and area-weighted DG methods are indistinguishable, therefore, only the meshes with the true volume DG method are shown. It can be observed that the true volume DG method gives good mesh evolution even though it is not a symmetry-preserving method.
the initial condition. Then pi , po , ρi and ρo denote the pressure and density at the inner and outer surface. The initial density, velocity, and pressure are given as follows, 1 (r 0 )2 −r 2 r 2 −(ri0 )2 0 γ−1 γ−1 ρ0 (r) = ( (r0 )o2 −(r0 )2 (ρ0i )γ−1 + (r0 )2 −(r ) , u 0 (r) = 0, p0 (r) = S 0 ∗ (ρ0 (r))γ . 0 )2 (ρo ) o o i i The analytic solution for the density, velocity, and pressure as a function of the spherical radius and time is given by, −2γ −2 ρ(r, t) = ρ0 (r0 )h(t) γ−1 , u(r, t) = r0 dh(t) p(r, t) = p0 (r0 )h(t) γ−1 . dt , Here, the spherical radius varies with time r(t) = h(t)r0 , and h(t) is defined by q h(t) = 1 − ( ttf )2 , r (ro0 )2 −(ri0 )2 where tf is the focusing time of the shell that is given by tf = γ−1 2 (c0o )2 −(c0i )2 with the sound speed defined q by c = γ ρp . In this paper, we use γ = 53 , ro0 = 1.0, ri0 = 0.9, ρ0o = 10−2 , ρ0i = 6.31 × 10−4 , p0o = 10, p0i = 0.1, S 0 = 2.15 × 104 and tf = 6.72 × 10−3 . In order to verify the accuracy of the DG(P1) methods, two uniformly refined meshes (i.e., 20 × 10 and 40 × 20 √) are used for a convergence study. Figure 6 presents the meshes from the two DG methods (20 × 10) at t = 23 tf . The mesh with the two DG methods are moving in a stable manner. The scatter plots of the √ density and pressure distribution at t = 23 tf are shown in Figure 7 and Figure 8 respectively. The density and pressure converge towards the analytical solution as expected with mesh refinement. In the scatter plots, compared with the area-weighted DG method (Figure ??), there are multiple values at the same radius with the results from the true volume DG method (Figure ??) demonstrating the symmetry errors. The accuracy and robustness of the area-weighted DG method is further demonstrated by calculating this test problem to t = 0.99tf . The mesh at t = 0.99tf is shown in Figure 9. From Figure 10, both variables converge toward the analytical solution with mesh refined. It is worth mentioning that this case can not run to t = 0.99tf with the true volume DG method because the true volume one is not symmetry preserving which is crucial for the robustness of this case.
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Table 1: The numerical error and convergence order of accuracy is shown for the free expansion test problem using the true volume DG(P1) method at t=0.4. Second-order accuracy is achieved. Density ρ
Mesh(K × L)
L2 error 6.6706e-5 1.7926e-5 4.6248e-6 1.1879e-6
10 × 5 20 × 10 40 × 20 80 × 40
Momentum ρu
order 1.90 1.96 1.96
L2 error 8.2750e-5 2.0208e-5 5.0146e-6 1.2396e-6
Pressure p
order 2.04 2.01 2.02
L2 error 1.2156e-4 3.2133e-5 7.9214e-6 1.9748e-6
order 1.93 2.02 2.01
Table 2: The numerical error and convergence order of accuracy is shown for the free expansion test problem using the area-weighted DG(P1) method at t=0.4. Second-order accuracy is achieved. Density ρ
Mesh(K × L)
L2 error 1.6109e-4 3.5231e-5 8.3853e-6 2.1609e-6
10 × 5 20 × 10 40 × 20 80 × 40
Momentum ρu
order 2.20 2.07 1.96
L2 error 1.1173e-4 3.0006e-5 7.0566e-6 1.7576e-6
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
Pressure p
order 1.90 2.09 2.01
L2 error 1.6048e-4 3.4037e-5 7.6103e-6 1.8921e-6
order 2.24 2.17 2.01
0.3
R
R
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.05
0.1
0.15
0.2
0.25
Z
0.3
0.35
0.4
0.45
0.5
Z
(a) The true volume method
(b) The area-weighted method √
Figure 6: The meshes are shown for the Kidder shell problem at t = area-weighted DG(P1) methods.
C.
3 2 tf
using the true volume and
Coggeshall test
The Coggeshall test problem27, 41 is a special case of a general set of solutions to the inviscid hydrodynamic equations in three dimensions from.13 It is a simple adiabatic compression problem that becomes singular at t = 1.0. A sphere of radius 1.0 is again filled with a perfect gas with γ = 5/3, and is collapsing with cylindrical (but not spherical) symmetry so, that the exact solution at point (R, Z) at a given time t is,
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0.1
0.1 DG(P1)+AW,20x10 DG(P1)+AW,40x20 Exact
0.08
0.08
0.06
0.06
Density
Density
DG(P1)+TV,20x10 DG(P1)+TV,40x20 Exact
0.04
0.04
0.02
0.02
0 0.45
0.46
0.47
0.48 Radius
0.49
0 0.45
0.5
0.46
(a)
0.48 Radius
0.49
0.5
0.498 Radius
0.499
0.5
(b)
0.1
0.1 DG(P1)+TV,20x10 DG(P1)+TV,40x20 Exact
DG(P1)+AW,20x10 DG(P1)+AW,40x20 Exact
0.08
0.08
0.06
0.06
Density
Density
0.47
0.04
0.04
0.02
0.02
0 0.495
0.496
0.497
0.498 Radius
0.499
0.5
0 0.495
0.496
(c)
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Figure 7: The scatter plots of density field are shown for the Kidder shell problem at t = 23 tf using the true volume and area-weighted DG(P1) methods. The exact solution is the solid black line. (a) The true volume method, (b) The area-weighted method, (c) The close-up with the true volume method and (d) The close-up with the area-weighted method. For the true volume DG method (the left figure a and c), there are multiple values at the same radius, which shows that this method is not symmetry-preserving. In contrast, the area-weighted DG method (the right figure b and d) preserves symmetry perfectly.
9
−R −Z 3Z ρ(t) = (1 − t)− 4 , u(t) = 1−t , v(t) = 4(1−t) , e(t) = ( 8(1−t) )2 . At the outer boundary, exact solution is prescribed while the boundary at R = 0 and Z = 0 are symmetric. In order to verify the accuracy of the DG(P1) method, a set of uniformly refined meshes (i.e., 9 × 10, 18 × 20, 36 × 40 and 72 × 80 ) are used for a convergence study. This case is only run using the area-weighted DG method. Figure 11 presents the mesh (9 × 10) at t = 0.60. The mesh is moving in a stable manner. The scatter plots of the density distribution at t = 0.6 are shown in Figure 12. The density converges towards the analytical solution as expected with mesh refinement.
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Figure 8: The scatter plots of pressure field is shown for the Kidder shell problem at t = volume and area-weighted DG(P1) methods. The exact solution is the solid black line.
3 2 tf
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Figure 9: The mesh is shown for the Kidder shell problem at t = 0.99tf using the area-weighted DG(P1) method. The mesh is moving perfectly in the radial direction.
D.
Spherical Sod problem
The spherical Sod shock tube40 is a classical test problem involving an ideal gas with γ = 75 . In this test problem, a spherical shock is created by a contact discontinuity that initially located at r = 0.5. The initial inner state is defined by (ρ0 , u0r , u0θ , p0 )L = (1, 0, 0, 1), and the initial outer state is defined by (ρ0 , u0r , u0θ , p0 )R = (0.125, 0, 0, 0.1). The spherical radial and angular components of the velocity are ur and uθ . This test case is run to t = 0.2. The calculations in this subsection are performed using a full polar mesh. The resolutions of the full polar mesh are 100 × 20. The outer boundary is a wall boundary condition. The calculations with both
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Figure 10: The results are shown for the Kidder shell problem at t = 0.99tf using the area-weighted DG(P1) method. The exact solution is the solid black line. The numerical results converge to the exact solution with mesh refinement.
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Figure 11: The mesh and density contours are shown for the Coggeshall problem at t = 0.6 using the areaweighted DG(P1) method. The mesh is moving in a table manner. The large density error occurs near the boundary Z = 0.
DG methods use limiters on the respective fields. The meshes at the final time for the true volume and area-weighted DG methods are shown in Fig. 13. The scatter plots of the density versus radius at the final time are shown in Fig. 14 for DG(P1). From Fig. 14c, the true volume method is not symmetry-preserving confirmed by the obvious density scatter distribution of the second-order DG(P1) with a full polar mesh while the area-weighted method is symmetry-preserving. E.
Implosion problem of Lazarus
This problem has an analytical solution that has been studied in other works including.5, 18 A sphere with an initial unit radius, a unit density, and a very small internal energy is driven by an inward spherical radial velocity specified at the outer boundary. The material is an ideal gas with γ = 5/3. The initial density, boundary velocity and internal energy are given as follows, ρ0 = 1, u0r = − (1−fαf u0θ = 0, e0 (r) = 10−5 . t)1−α , Here, α = 0.6883545, f = 1 − εt − δt3 , ε = 0.185 and δ = 0.28. It can be observed that this spherical radial velocity at the outer boundary is time-varying and calculated from a self-similar solution.5 The shock
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9 DG(P1)+AW, 9x10 DG(P1)+AW,18x20 DG(P1)+AW,36x40 DG(P1)+AW,72x80 Exact
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Figure 12: The scatter plots of the density are shown for the Coggeshall problem at t = 0.6 using the area-weighted DG(P1) method. With mesh refined, the density distribution converges toward the analytical solution.
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Figure 13: The final meshes are shown for the spherical Sod shock problem at t = 0.20 using the DG(P1) methods. (a) The true volume method with a full polar mesh, (b) The area-weighted method with a full polar mesh a, (c) The close-up with the true volume method on a full polar mesh and (d) The close-up with the area-weighted method on a full polar mesh. Compared with the right panel (d) (the area-weighted DG method), the mesh (c) is not aligned well with the radial direction since the true volume DG method is not symmetry preserving.
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Figure 14: The density scatter plots of the spherical Sod shock problem at t = 0.20 using a 100 radial elements with DG(P1). The reference solution is the solid black line. (a) The true volume method with a full polar mesh, (b) The area-weighted method with a full polar mesh, (c) The close-up with the true volume method on a full polar mesh and (d) The close-up with the area-weighted method on a full polar mesh. Compared with the right panel (the area-weighted DG method), there are multiple values at the same radius (the left panel) showing that the true volume DG method is not symmetry preserving.
converges at the origin of the sphere at t = 0.75 and the density is flat at about 9.5. At a time of t = 0.8, the outward propagating radial shock arrives at a radius of about 0.11 with a density peak of 31.5. For this test problem, the computational domain is defined in polar coordinates. Both a partial and full polar mesh are used for calculations. The resolutions of the partial and full polar mesh are 100 × 20 and 100×40 respectively. The calculations with the true volume DG method use the partial polar mesh while the calculations with the area-weighted DG use the full one. Limiters are used with DG(P1) for all calculations of this test problem. The meshes at t = 0.80, using the true volume and area-weighted DG methods, are shown in Fig. 15. From the enlarged view near the origin in Fig. 15c, the mesh is not perfectly aligned with the radial direction with the true volume method. In contrast, the area-weighted DG method preserves the spherical symmetry as shown in Fig. 15d. Scatter plots of the density versus radius at t = 0.74 and t = 0.80 are shown in Fig. 16. Multiple density values are present at the same position, e.g. the layer near the origin, in the results at t = 0.80 (16a ) since the true volume method is not symmetry-preserving. Over all, the numerical results from both methods agree favorably with the reference solution.9 In addition, at t = 0.74, the true volume DG method captures the shock position more accurately compared with the area-weighted DG method. The area-weighted DG method does not conserve the physical momentum and physical total energy with a finite mesh size.
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Figure 15: The final meshes are shown for the implosion problem of Lazarus at t = 0.80 using DG(P1). (a) The global view using the true volume DG method on a partial polar mesh, (b) The global view using the area-weighted DG method on a full polar mesh, (c) The close-up near the origin with the true volume method on a partial polar mesh, (d) The close-up near the origin with the area-weighted method on a full polar mesh. From figure (c), the mesh near the origin is not aligned well with the radial direction since the true volume method is not symmetry-preserving. From figure (d), the area-weighted method preserves spherical symmetry on this test case, so the mesh is aligned perfectly with the radial direction.
F.
Sedov blast wave problem
The Sedov blast wave test problem37, 38 is an outward traveling blast wave in a gamma-law gas that is initiated by an energy source. In this work we use γ = 7/5. The initial conditions are given by (ρ0 , u0r , u0θ , p0 ) = (1.0, 0, 0, 10−6 ) and there is an energy source at the origin. The pressure in the elements at the origin is o given by po = (γ − 1)ρo E Vo , where Vo denotes the combined volume of the elements in the origin region and Eo is the total amount of released internal energy. The energy source is selected to be equal to 0.425536 so that the shock front is located at a radius equal to 1 at t = 1. This test problem is calculated using box meshes. The computational domain is defined in 2D Cartesian coordinates [0, 1.2]×[0, 1.2]. The mesh resolutions is 30×30 uniformly which is used for the calculations with the true volume and area-weighted DG methods. The center element contains the energy source. Symmetric boundary conditions are applied to the left and bottom boundaries. The other boundaries are fixed. The calculations are performed with the DG(P1) method with limiters. The velocity limiting is performed in
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Figure 16: The scatter plots of density using DG(P1) are shown for the implosion problem of Lazarus using a partial polar grid with the true volume method and a quadrant of a full polar grid with the area-weighted method. (a) The true volume method with a partial polar mesh, (b) The true volume method with a full polar mesh and (c) The area-weighted method with a full polar mesh. The density distributions with both the true volume and area-weighted methods agree well with the reference solution. At t = 0.74, the true volume method (the left figure) captures the shock position more accurately compared to the area-weighted method (the right figure).
the local flow direction. Fig. 17 shows the deformed meshes at t = 1.0 with both the true volume and area-weighted DG methods. The scatter plots of the density versus radius at the final time are shown in Fig. 18. The error in shock position with the area-weighted method is likely due to the fact that this method doesn’t conserve the physical momentum and physical total energy with a finite mesh size.
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Figure 17: The final meshes using DG(P1) are shown for the Sedov blast problem on a box grid at t = 1.0. The resolution is 30x30 elements. The meshes are deforming in a stable and smooth manner that is consistent with the analytic solution.
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Figure 18: The scatter plots of the density using DG(P1) are shown for the Sedov problem on the box mesh. Compared with the area-weighted method, the true volume method captures the shock more accurately.
G.
Noh problem
The Noh test problem36 is commonly used to demonstrate the robustness of Lagrangian methods. The material in this test problem is an ideal gas with γ = 5/3. The initial velocity is directed radially inward with a magnitude of 1, the density is unity, and the pressure is given by p0 = 10−6 . The converging flow causes a shock to form at the origin and it propagates radially outward. The density plateau behind the shock wave reaches 64. The computational domain is defined in Cartesian coordinates [0.0, 1.0] × [0.0, 1.0]. The mesh resolutions used for the Noh calculations is 50 × 50 uniformly. Symmetric boundary conditions are applied to the left and bottom boundaries. The boundary condition on the outer boundary surface is a constant pressure of pb = 10−6 . The calculations are performed with the DG(P1) method with limiters. The velocity limiting is performed in the principle strain direction. Fig. 19 shows the deformed meshes (50 × 50) near the origin at the final time with both the true volume and area-weighted methods. From Fig. 19a, the mesh with the true volume DG method looks reasonable except for the 2 layers of elements near the Z axis. The scatter plots of the density versus radius at the final time are shown in Fig. 20. In contrast, compared with Fig. 20a, the results shown in Fig. 20b demonstrate that the area-weighted DG method is good agreement with the analytic solution and has far better spherical symmetry on a box mesh than the true volume method.
VIII.
Conclusions
In this paper, we presented both true volume and area-weighted DG hydrodynamic methods for unstructured meshes in axisymmetric coordinates. The physical evolution equations (velocity, and specific total energy) are discretized using the DG method and the modal fields are approximated with linear Taylor expansions on a reference element while the density is advanced by a modal method about the centroid of the reference element . The true volume approach discretizes the governing equations using the 3D volume. In contrast, the area-weighted approach approximates the integrals as a 2D Cartesian surface rotated about the axis of symmetry using a element average radius. Different multidirectional approximate Riemann problems at the element surface nodes are solved for these two approaches. The true volume DG method conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve the spherical symmetry on an equal-angle polar grid for a 1D radial flow. The area-weighted DG method conserves mass and it preserves spherical symmetry on an equal-angle polar grid for a 1D radial flow, but it violates the GCL. The area weighted DG method only conserves the physical momentum and physical total energy in the infinitesimal mesh size. The accuracy and robustness of the new Lagrangian DG hydrodynamic methods were demonstrated by calculating a suite of challenging test problems. The free expansion problem and the Kidder shell problem are smooth flow test cases and were used to demonstrate that both the true volume and area-weighted
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Figure 19: The final meshes near the origin using DG(P1) are shown for the Noh problem on a box mesh at t = 0.6 using a mesh resolution of 50 × 50. Compared with the area-weighted method (the right figure), there are two layers near the Z axis (the horizontal axis) that move badly with the true volume method (the left figure).
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Figure 20: The scatter plots of density using DG(P1) are shown for the Noh problem on a box mesh at t = 0.6. The true volume method (the left figure) generates large density errors near the Z axis; in contrast, the density field with the area-weighted method (the right figure) is less scattered.
DG methods can deliver approximately second-order accuracy. The spherical Sod problem, the Lazarus implosion problem, the Sedov blast wave problem, and the Noh problem have strong shocks and were used to demonstrate the accuracy and robustness of the methods. The Lagrangian DG methods for axisymmetric coordinates presented in the paper show great promise for simulating both smooth flows, and flows with discontinuities and shocks.
IX.
Acknowledgments
We gratefully acknowledge the support of the NNSA through the Laboratory Directed Research and Development (LDRD) program at Los Alamos National Laboratory. The Los Alamos unlimited release
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number is LA-UR-17-30310. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC for the U.S. Department of Energy NNSA under Contract No. DE-AC52-06NA25396.
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