Variational Reconstruction

• Variational Reconstruction
  • Functional Design:
    • Selecting Inner Product
    • Other Forms
  • Reconstruction Matrices

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Reconstruction from mean values with piecewise zero-mean basis:

$$ u_i(\mathbf{x}) =\overline{u}_i +\sum_{l=1}^{N_{base}}{ u^l_i\varphi^l_i(\mathbf{x})} $$

Functional Design:

For face $f$ with cell on both sides $L,R$, reconstruction functional on the face is designed as:

$$ I_f=w_g(f)\int_f{d\Gamma \sum_{p=0}^k {w_d(p)^2 \left| \mathcal{D}_pu_L -\mathcal{D}_pu_R \right|_{\langle\rangle _{f,p}}^2}} $$ where

• $w_g$ is Geometric Weight, which could well be dimensioned. An original choice is to use $w_g=S_f^{-1}$, where $S_f$ is area of $f$.
• $w_d$ is Derivative Weight, which is defined to be dimensionless here.
• $\mathcal{D}_p u$ is a tensor-like (covariance only in linear transformation of space from , no curvilinear) variable representing the derivatives of $u$ , for example, the Cartesian components: $$ [\mathcal{D}_3 u]_{ijk} \equiv [u]_{x_jx_jx_k} \equiv \frac{\partial^3u}{ \partial x_i \partial x_j\partial x_k} $$
• $\left|\cdot\right|_{\langle\rangle _{f,p}}^2$ is defined with $$ \left|A\right|_{\langle\rangle _{f,p}}^2 =\langle A,A \rangle _{f,p} $$ where $A$ is a tensor of p_th order. The $\langle,\rangle$ is a inner product.

Selecting Inner Product

The Normal Functional from Wang:

$$ \langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3} =d_f^6 \partial_{nnn}u \partial_{nnn}v $$

X-Y aligned from Pan:

$$ \begin{aligned} \langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3} &=(\Delta_x^3 \partial_{xxx} u) (\Delta_x^3 \partial_{xxx} v)\ &+(3\Delta_x^2\Delta_y \partial_{xxy} u) (3\Delta_x^2\Delta_y \partial_{xxy} v)\ &+(3\Delta_x\Delta_y^2 \partial_{xyy} u) (3\Delta_x\Delta_y^2 \partial_{xyy} v)\ &+(\Delta_y^3 \partial_{yyy} u) (\Delta_y^3 \partial_{yyy} v)\ \end{aligned} $$

Pre-Isotropic from Huang:

$$ \begin{aligned} \langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3} &=(d_f^3 \partial_{xxx} u) (d_f^3 \partial_{xxx} v)\ &+(d_f^3 \partial_{xxy} u) (d_f^3 \partial_{xxy} v)\ &+(d_f^3 \partial_{xyy} u) (d_f^3 \partial_{xyy} v)\ &+(d_f^3 \partial_{yyy} u) (d_f^3 \partial_{yyy} v)\ \end{aligned} $$

Other Forms

Could be re-written to:

$$ I_f=w_g(f)\int_f{d\Gamma \sum_{p=0}^k { w_d(p)^2 L_f^{2p} \left| \mathcal{D}_pu_L -\mathcal{D}_pu_R \right|_{\langle\rangle _{f,p}}^2 } } $$

• $L_f$ is a facial length reference, all the anisotropy within derivative components are dealt in the inner product with a relative (dimensionless form). Output of inner product becomes dimensional.

Reconstruction Matrices

$$ \begin{aligned} A_{mn}^i & = \sum_{j\in S_i} { w_g(f_{i-j}) \int_{f_{i-j}}{\sum_{p=0}^k{ w_d(p)^2 \left\langle{\mathcal{D}_p(\varphi^m_i),\mathcal{D}_p(\varphi^n_i)}\right\rangle_{f_{i-j},p} } d\Gamma}} \ B_{mn}^{i-j} & = { w_g(f_{i-j}) \int_{f_{i-j}}{\sum_{p=0}^k{ w_d(p)^2 \left\langle{\mathcal{D}_p(\varphi^m_i),\mathcal{D}_p(\varphi^n_j)}\right\rangle_{f_{i-j},p} } d\Gamma}} \ b_{m}^{i-j} & = { w_g(f_{i-j}) \int_{f_{i-j}}{{ w_d(0)^2 \left\langle{\varphi^m_i,1}\right\rangle_{f_{i-j},p} } d\Gamma}} \end{aligned} $$

with the reconstruction system in local form:

$$ A_{mn}^i u_i^n = \sum_{j\in S_i}{ \left(B_{mn}^{i-j} u_j^n +b_{m}^{i-j}(\overline{u}_j - \overline{u}_i) \right)} $$