Internal Hierarchical Polynomial Model for Physics Simulation
A method is provided for using a hierarchical polynomial model for physics simulation. The method includes obtaining coupled equations for a physics simulation. The coupled equations have variables and boundary conditions that constrain the variables. The method also includes generating meshes corresponding to the coupled equations. The method iteratively solves for the boundary conditions that depend on the variables within the meshes to convergence to improve numerical stability of the physics simulation, including: (i) solving for each variable in a first mesh, while holding the other meshes to a weak convergence, to obtain a first solution; (ii) applying the first solution to resolve a second mesh, to obtain a second solution; and (iii) generating a hierarchical polynomial based on the second solution. The hierarchical polynomial is a functional form of a pre-determined physics equation. The method also (iv) computes new boundary conditions for resolving the meshes, using the hierarchical polynomial.
1 . A method of physics simulation, the method comprising:
obtaining a set of coupled physics equations for a physics simulation, wherein the set of coupled physics equations has (i) a set of variables and (ii) boundary conditions that constrain the set of variables;
obtaining a plurality of meshes corresponding to the set of coupled physics equations, wherein each mesh is a lattice of discrete points in a multidimensional space corresponding to a respective subset of variables of the set of variables; and
iteratively solving for the boundary conditions that implicitly depend on the set of variables within the plurality of meshes to convergence to improve numerical stability of the physics simulation, including:
solving for each variable in a first mesh of the plurality of meshes, while holding the other meshes to weak convergence criteria, to obtain a first solution;
applying a solution for one or more variables in the first solution to resolve a second mesh of the plurality of the meshes, to obtain a second solution;
generating a hierarchical polynomial based on the second solution, wherein the hierarchical polynomial (i) is a functional form of a pre-determined physics equation and (ii) includes one or more physics equations in an exponent; and
computing new boundary conditions for resolving the plurality of meshes, using the hierarchical polynomial.
2 . The method of claim 1 , wherein the set of coupled equations includes an equation for a solid temperature field and an equation for a fluid temperature field.
3 . The method of claim 1 , wherein the physics simulation minimizes a physics metric.
4 . The method of claim 3 , wherein the physics metric is a determination of flow rates for N fluid channels that equalize the pressure drop across all channels in which energy can be conducted away from a fluid channel, through a solid, and into another fluid channel.
5 . The method of claim 4 , wherein the new boundary conditions alter the flow rates for the N fluid channels to equalize pressure
6 . The method of claim 1 , wherein the physics simulation finds a pattern for a flow straightening device for a flow distribution to achieve isothermal exit conditions.
7 . The method of claim 6 , wherein the new boundary conditions alter the flow distribution.
8 . The method of claim 1 , wherein the physics simulation exhibits numerical instabilities.
9 . The method of claim 1 , wherein the plurality of meshes includes a Finite Element Method (FEM) mesh for a solid material region.
10 . The method of claim 1 , wherein the plurality of meshes includes a Finite-Difference Method (FDM) mesh for a fluid region.
11 . The method of claim 1 , wherein each mesh further corresponds to a respective region with distinct physical properties.
12 . The method of claim 1 , wherein iterating is performed using Picard iterations.
13 . The method of claim 1 , wherein:
the plurality of meshes includes a Finite Element Method (FEM) mesh for a solid material region and a Finite-Difference Method (FDM) mesh for a fluid region;
the FEM mesh is resolved independently of the FDM mesh that is held to weak convergence criteria;
the FDM mesh applies pertinent information from the FEM solution to the FDM mesh and resolves a step in the FDM physics field; and
prior to starting a next iteration in the FEM solution, the hierarchical polynomial is used to provide the new boundary conditions for resolving a physics metric.
14 . The method of claim 13 , wherein the set of coupled physics equations solve for temperature in a fluid and a solid of a two-pass exchanger to determine how the flow re-distributes in the exchanger so that pressure is equalized.
15 . The method of claim 14 , wherein:
a solution for the FEM mesh is defined by the equation:
∂
T
∂
t
=
a
·
∇
2
T
wherein T is temperature, t is time, and a is a constant.
16 . The method of claim 15 , wherein a solution for the FDM mesh is defined by the equations:
1
A
f
Δ
m
˙
Δ
z
Δ
t
-
m
˙
2
ρ
2
·
A
f
2
Δ
ρ
=
-
Δ
P
+
f
Δ
z
2
·
D
h
m
˙
2
ρ
A
f
2
A
f
ρ
c
p
Δ
z
Δ
t
Δ
T
+
c
p
m
˙
Δ
T
=
Q
wherein {dot over (m)} is a mass flow rate boundary condition.
17 . The method of claim 16 , wherein FDM results are used to compute the hierarchical polynomial by applying the following equation on each fluid channel:
m
˙
i
+
1
=
m
˙
i
·
sgn
(
P
T
-
P
i
)
(
❘
"\[LeftBracketingBar]"
P
i
P
T
-
1
❘
"\[RightBracketingBar]"
)
C
0
·
m
.
i
wherein P T = P , i is a current iteration value, i+1 is a next iteration value, P T is a target pressure drop, P is average pressure drop of all channels, and sgn is the numerical sign.
18 . A computer system for physics simulation, comprising:
one or more processors; and
memory;
wherein the memory stores one or more programs configured for execution by the one or more processors, and the one or more programs comprise instructions for performing the method of claim 1 .
19 . A computer system for physics simulation, comprising:
one or more processors; and
memory;
wherein the memory stores one or more programs configured for execution by the one or more processors, and the one or more programs comprise instructions for performing the method of claim 17 .
20 . A non-transitory computer readable storage medium storing one or more programs configured for execution by a computer system having one or more processors and memory, the one or more programs comprising instructions for performing the method of claim 1 .
21 . A non-transitory computer readable storage medium storing one or more programs configured for execution by a computer system having one or more processors and memory, the one or more programs comprising instructions for performing the method of claim 17 .