Computation of shape functions - Revision history

E6WikiAdmin at 17:02, 4 December 2022

2022-12-04T17:02:20Z

E6WikiAdmin at 15:28, 10 November 2022

2022-11-10T15:28:49Z

AndrejKP: Typos

2022-06-16T12:58:39Z

Typos

NikaMlinaric: Added missing $\mathcal{L}$ in the 3rd equation of Derivation

2022-03-03T09:32:59Z

Added missing $\mathcal{L}$ in the 3rd equation of Derivation

Jureslak: /* Another view on derivation of shape functions */

2020-08-29T09:26:34Z

‎Another view on derivation of shape functions

Jureslak: /* Another view on derivation of shape functions */

2020-08-29T09:24:41Z

‎Another view on derivation of shape functions

Gkosec: /* Another view on derivation of shape functions */

2019-10-28T17:00:06Z

‎Another view on derivation of shape functions

Jureslak: /* Numerical calculation of the shape functions */

2019-09-08T17:59:22Z

‎Numerical calculation of the shape functions

Jureslak at 17:58, 8 September 2019

2019-09-08T17:58:42Z

Jureslak: /* Numerical calculation of the shape functions */

2019-09-08T17:58:21Z

‎Numerical calculation of the shape functions

@@ Line 2: / Line 2: @@
 [https://e6.ijs.si/ParallelAndDistributedSystems/publications/69777155.pdf J. Slak, G. Kosec; Medusa : A C++ library for solving PDEs using strong form mesh-free methods, ACM transactions on mathematical software, vol. 47, 2021 [DOI: 10.1145/3450966]]
 }}
 Shape functions, often called stencil weights, are functionals, approximating linear partial differential operators at a chosen point.

← Older revision		Revision as of 15:28, 10 November 2022
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		+	{{Box-round\|title= Related papers \|
		+	[https://e6.ijs.si/ParallelAndDistributedSystems/publications/69777155.pdf J. Slak, G. Kosec; Medusa : A C++ library for solving PDEs using strong form mesh-free methods, ACM transactions on mathematical software, vol. 47, 2021 [DOI: 10.1145/3450966]]
		+	}}
		+
	Shape functions, often called stencil weights, are functionals, approximating linear partial differential operators at a chosen point.		Shape functions, often called stencil weights, are functionals, approximating linear partial differential operators at a chosen point.
	A shape function $\b{\chi}_{\mathcal{L}} (\vec{p})$ for a linear partial differential operator $\mathcal{L}$		A shape function $\b{\chi}_{\mathcal{L}} (\vec{p})$ for a linear partial differential operator $\mathcal{L}$

@@ Line 1: / Line 1: @@
 Shape functions, often called stencil weights, are functionals, approximating linear partial differential operators at a chosen point.
-A shape function $\b{\chi}_{\mathcal{L}} (\vec{p})$ for a linear parial differential operator $\mathcal{L}$
+A shape function $\b{\chi}_{\mathcal{L}} (\vec{p})$ for a linear partial differential operator $\mathcal{L}$
 in a point $p$ is a vector of stencil weights, such that
 \[
@@ Line 49: / Line 49: @@
 \[ u''(s_2) \approx \underbrace{\begin{bmatrix} \frac{1}{h^2} & \frac{-2}{h^2} & \frac{1}{h^2} \end{bmatrix}}_{\b{\chi}^\T} \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix}  \]
-The fact that $\b{\chi}$ is independent of the function values $\b{u}$ but depend only on domain geometry means that
+The fact that $\b{\chi}$ is independent of the function values $\b{u}$ but depends only on domain geometry means that
 '''we can just compute the shape functions $\b{\chi}$ for points of interest and then approximate any linear operator
 of any function, given its values, very fast, using only a single dot product.'''

@@ Line 23: / Line 23: @@
 and applying operator $\mathcal{L}$ we get
 \[
-     \hat u (\vec{p}) = (\mathcal{L}\b{b})(\vec{p})^\T \b{\alpha} =
+     \mathcal{L} \hat u (\vec{p}) = (\mathcal{L}\b{b})(\vec{p})^\T \b{\alpha} =
 (\mathcal{L}\b{b})(\vec{p})^\T (B^\T W^2 B)^{-1} B^\T W^2 \b{u}
 = \b{\chi}_{\mathcal{L}}(\vec{p})^\T \b{u}

@@ Line 110: / Line 110: @@
 Using the solution derived in [[Solving underdetermined systems]] and substituting inverse weights $W = W^{-1}$ and matrix $A = B^\T$, we get the solution
 $$
-\b{\chi}_{\mathcal{L}} (\vec{p}) = (W^{-1})^{-2} (B^\T)^\T (B^\T (W^{-1})^{-2}(B^\T)^\T) (\mathcal{L}\b{b})(\vec p) = W^2 B (B^\T W^2 B)^{-1} (\mathcal{L}\b{b})(\vec p).
+\b{\chi}_{\mathcal{L}} (\vec{p}) = (W^{-1})^{-2} (B^\T)^\T (B^\T (W^{-1})^{-2}(B^\T)^\T)^{-1} (\mathcal{L}\b{b})(\vec p) = W^2 B (B^\T W^2 B)^{-1} (\mathcal{L}\b{b})(\vec p).
 $$
 After transposing, we get

@@ Line 90: / Line 90: @@
   \chi_{\mathcal{L}, 1} \\ \vdots \\ \chi_{\mathcal{L}, n}
 \end{bmatrix}
 \begin{bmatrix}
    (\mathcal{L} b_1)(\vec{p}) \\ \vdots \\ (\mathcal{L} b_m)(\vec{p})

@@ Line 132: / Line 132: @@
 In general, the system $B^\T\b{\chi}_{\mathcal{L}}(\vec{p}) = (\mathcal{L}\b{b})(\vec p)$ is underdetermined and must be solved in the least-weighted-norm sense.
 This means solving the underdetermined system $B^\T W y =  (\mathcal{L}\b{b})(\vec p)$ and computing $\b{\chi}_{\mathcal{L}}(\vec{p}) = Wy$.
-For more details on this, see [[Solving underdeternimed systems]].
+For more details on this, see [[Solving underdetermined systems]].
 Depending on our knowledge of the

@@ Line 124: / Line 124: @@
 The alternative view above is useful in this aspect.
-=== Invertible $B$ ==
+=== Invertible $B$ ===
 If $B$ is invertible, then $\b{\chi}_{\mathcal{L}}(\vec{p})$ can be thought as a solution of a system $B^\T\b{\chi}_{\mathcal{L}}(\vec{p}) = (\mathcal{L}\b{b})(\vec p)$, which can be solved using LU or Cholesky decomposition for example.
 For more on this, see [[Solving linear systems]].
-=== General case ==
+=== General case ===
 In general, the system $B^\T\b{\chi}_{\mathcal{L}}(\vec{p}) = (\mathcal{L}\b{b})(\vec p)$ is underdetermined and must be solved in the least-weighted-norm sense.
 This means solving the underdetermined system $B^\T W y =  (\mathcal{L}\b{b})(\vec p)$ and computing $\b{\chi}_{\mathcal{L}}(\vec{p}) = Wy$.

@@ Line 122: / Line 122: @@
 == Numerical calculation of the shape functions ==
 Numerically cheaper and more stable way is to translate the problem of inverting the matrix to solving a linear system of equations.
-'''Invertible $B$ case:'''
+=== Invertible $B$ ==
-'''General case:'''
+If $B$ is invertible, then $\b{\chi}_{\mathcal{L}}(\vec{p})$ can be thought as a solution of a system $B^\T\b{\chi}_{\mathcal{L}}(\vec{p}) = (\mathcal{L}\b{b})(\vec p)$, which can be solved using LU or Cholesky decomposition for example.
-For a system written as $Ax = b$, where $A$ is a $n\times m$ matrix, $x$ is a vector of length $m$ and $b$ a vector of length $n$, a generalized solution
+For more on this, see [[Solving linear systems]].
-In our case, let us denote a part of the solution containing the pseudoinverse by $\tilde{\b{\chi}}_{\mathcal{L}, \vec{p}}$.
+=== General case ==
-\[ \b{\chi}_{\mathcal{L}, \vec{p}}(\vec{p}) = \underbrace{(\mathcal{L}\b{b})(\vec{p})^\T (WB)^+}_{\tilde{\b{\chi}}_{\mathcal{L}, \vec{p}}} W \]
+In general, the system $B^\T\b{\chi}_{\mathcal{L}}(\vec{p}) = (\mathcal{L}\b{b})(\vec p)$ is underdetermined and must be solved in the least-weighted-norm sense.
-We have an expression $\tilde{\b{\chi}}_{\mathcal{L}, \vec{p}} = (\mathcal{L}\b{b})(\vec{p})^\T (WB)^+$ which after transposition takes the form
+This means solving the underdetermined system $B^\T W y =  (\mathcal{L}\b{b})(\vec p)$ and computing $\b{\chi}_{\mathcal{L}}(\vec{p}) = Wy$.
-$\tilde{\b{\chi}}_{\mathcal{L}, \vec{p}}^\T = ((WB)^\T)^+(\mathcal{L}\b{b})(\vec{p})$, the same as $x = A^+b$ above.
+For more details on this, see [[Solving underdeternimed systems]].
-The system before can be solved using any decomposition of matrix $(WB)^\T = B^\T W$, not neceserally the SVD decompostion, but depending on our knowledge of the
+Depending on our knowledge of the
 problem, we can use Cholesky ($B^\T W$ is positive definite), $LDL^\T$ if it is symmetric, $LU$ for a general square matrix, $QR$ for full rank overdetermined system and SVD for a general system.
 If more shapes need to be calculated using the same matrix $B^\T W$ and only different right hand sides, it can be done efficiently by storing the decomposition of $B^\T W$.
 == Linearity and reduction to only computing the shapes of the operator space basis ==