Residual Norms

During the numerical solution, Simcenter STAR-CCM+ allows you to monitor the residual vector, the solution increment, and the energy increment by means of different norms.

At every iteration i, Simcenter STAR-CCM+ computes the solution update ${Δ u}^{i} \in ℝ^{N}$ by solving the linear system of equations:

Figure 1. EQUATION_DISPLAY

K^{i} {Δ u}^{i} = r^{i}

(4833)

with Newton iterations. $\begin{matrix} K^{i} \in ℝ^{N} \times ℝ^{N} \end{matrix}$ and $r^{i} \in ℝ^{N}$ denote the material tangent matrix and the residual vector at the i-th iteration, respectively, with $N$ being the size of the system (that is, the total number of degrees of freedom).

For the Solid Stress solver, $r^{i}$ and ${Δ u}^{i}$ denote the residual force and the displacement increment. For the Finite Element Solid Energy solver, they indicate the residual flux and the temperature increment. For the finite element Magnetic Vector Potential solver, they indicate the residual (which is related to the electric current load) and the magnetic vector potential increment.

After each iteration, the solvers compute the norms of the residual ( $r^{i}$ ), the variable increment ( ${Δ u}^{i}$ ), and the energy increment ( $W^{i}$ ). When specified, these norms are normalized to the norm-value calculated for the 1st iteration:

Figure 2. EQUATION_DISPLAY

\begin{matrix} \begin{matrix} \begin{matrix} \bar{r^{i}} = \frac{‖ r^{i} ‖}{‖ r^{1} ‖}; \end{matrix} & \bar{{Δ u}^{i}} = \frac{‖ {Δ u}^{i} ‖}{‖ {Δ u}^{1} ‖}; & \bar{W^{i}} = \frac{W^{i}}{W^{1}} \end{matrix} \end{matrix}

(4834)

The norms of the residual vector, $‖ r^{i} ‖$ , and the variable increment vector, $‖ {Δ u}^{i} ‖$ , can be calculated using different methods (see Residual and Variable Increment Norms), whereas the energy increment $W^{i}$ is calculated as the inner product between the variable increment vector and the residual vector (see Energy Increment). In solid stress applications, $W^{i}$ is called the strain energy; in solid energy applications, $W^{i}$ is called the thermal potential; in electromagnetic applications, $W^{i}$ is the magnetic energy increment.

Residual and Variable Increment Norms

Simcenter STAR-CCM+ allows you to monitor the residual and the primary variable increment by means of three different norms. To simplify the notation, the following definitions of norm consider a general vector $a$ that represents either the variable increment ${Δ u}^{i}$ or the residual $r^{i}$ at the i-th iteration.

The components of $a$ are denoted by $a_{j}$ , with $j = 1, ..., N$ , where $N$ is the vector dimension (that is, the total number of degrees of freedom of the system, Eqn. (4833)).

Maximum Norm: The maximum norm of a vector $a$ is the maximum of the absolute values of its components, $a_{j}$ :
Figure 3. EQUATION_DISPLAY

${‖ a ‖}_{\max} = \max_{j = 1, ..., N} (| a_{j} |)$
(4835)
L¹-norm: The L¹ norm of a vector $a$ is the arithmetic mean of the absolute values of its components, $a_{j}$ :
Figure 4. EQUATION_DISPLAY

${‖ a ‖}_{L^{1}} = \frac{1}{N} \sum_{j = 1}^{N} | a_{j} |$
(4836)
Euclidean Norm: The Euclidean norm, or L²-norm, of a vector $a$ is the Euclidean length of $a$ normalized to the number of its components:
Figure 5. EQUATION_DISPLAY

${‖ a ‖}_{L^{2}} = \frac{1}{N} \sqrt{\sum_{j = 1}^{N} a_{j}^{2}}$
(4837)

Energy Increment

Simcenter STAR-CCM+ computes the energy increment added in one iteration of the Newton scheme as:

Figure 6. EQUATION_DISPLAY

W^{i} = \frac{1}{2} \sum_{j = 1}^{N} {Δ u}_{j}^{i} r_{j}^{i}

(4838)

where ${Δ u}_{j}^{i}$ and $r_{j}^{i}$ are the components of the variable increment vector and the residual vector at the i-th iteration, and $N$ is the vector dimension.