## 3D Surface Plot: True Error vs. x1 and x2 ### Overview The image is a 3D surface plot visualizing the difference between a "True" value and a "FE" (likely Finite Element) approximation as a function of two variables, x1 and x2. The surface is colored, with the color gradient indicating the magnitude of the difference. The plot shows how the error changes across the x1-x2 plane. ### Components/Axes * **X-axis:** Labeled "x1". Scale ranges from 0.0 to 1.0 in increments of 0.2. * **Y-axis:** Labeled "x2". Scale ranges from 0.0 to 1.0 in increments of 0.2. * **Z-axis:** Labeled "True α - FE". Scale ranges from 0.0 to 2.5 in increments of 0.5. * **Surface Color:** The surface color varies from dark blue/purple at the origin to yellow at the top-right corner, indicating increasing values of "True α - FE". ### Detailed Analysis * **Surface Trend:** The surface starts near zero at the origin (x1=0, x2=0) and rises sharply as both x1 and x2 increase. The rate of increase appears to be non-linear, possibly quadratic or exponential. * **x1 = 0 Slice:** Along the x2 axis (x1=0), the "True α - FE" value remains close to zero for all values of x2. * **x2 = 0 Slice:** Along the x1 axis (x2=0), the "True α - FE" value remains close to zero for all values of x1. * **x1 = 1, x2 = 1:** At the point (x1=1, x2=1), the "True α - FE" value reaches its maximum, approximately 2.7. * **Linear Projections:** There are dark purple lines projected onto the x1-x2 plane. These lines originate from the surface and extend down to the x1-x2 plane. They appear to be parallel and evenly spaced, providing a visual aid for understanding the surface's shape. ### Key Observations * The error ("True α - FE") is minimal when either x1 or x2 is close to zero. * The error increases significantly as both x1 and x2 approach 1. * The surface exhibits a non-linear relationship between x1, x2, and the error. ### Interpretation The plot suggests that the Finite Element (FE) approximation is most accurate when either x1 or x2 is small. As both x1 and x2 increase, the FE approximation deviates significantly from the "True" value, leading to a substantial error. This could indicate that the FE model is less accurate in regions where both x1 and x2 are large, possibly due to boundary effects, mesh density, or other factors related to the FE method. The non-linear relationship suggests that the error is not simply additive with respect to x1 and x2, but rather a more complex interaction between them. The parallel lines projected onto the x1-x2 plane are a visual aid to help understand the slope of the surface.

\n ## 3D Surface Plot: True ∝ FE ### Overview The image depicts a 3D surface plot representing a relationship between three variables: `x₁`, `x₂`, and `True ∝ FE`. The surface is colored with a gradient, transitioning from dark purple to teal to yellow, indicating varying values of `True ∝ FE`. The plot appears to visualize a function where the value of `True ∝ FE` increases as both `x₁` and `x₂` increase, but the relationship is not strictly linear. ### Components/Axes * **X-axis:** Labeled `x₁`, ranging from approximately 0.0 to 1.0. * **Y-axis:** Labeled `x₂`, ranging from approximately 0.0 to 1.0. * **Z-axis:** Labeled `True ∝ FE`, ranging from approximately 0.0 to 2.5. * **Surface:** The colored surface represents the function being plotted. The color gradient indicates the value of `True ∝ FE`. * **Grid:** A 3D grid is present, providing a visual reference for the axes and surface. ### Detailed Analysis The surface starts at a low value of `True ∝ FE` (approximately 0.0) when either `x₁` or `x₂` is close to 0.0. As both `x₁` and `x₂` increase, the value of `True ∝ FE` rises. The maximum value of `True ∝ FE` (approximately 2.5) is reached when both `x₁` and `x₂` are close to 1.0. The surface is relatively flat near the axes (where either `x₁` or `x₂` is close to 0.0) and becomes steeper as both variables approach 1.0. The color gradient shows: * Dark purple: `True ∝ FE` values around 0.0 - 0.5 * Blue: `True ∝ FE` values around 0.5 - 1.5 * Teal: `True ∝ FE` values around 1.5 - 2.0 * Yellow: `True ∝ FE` values around 2.0 - 2.5 The surface appears to be smooth and continuous, suggesting that the function is well-defined within the plotted range. There are no visible discontinuities or sharp edges. ### Key Observations * The value of `True ∝ FE` is positively correlated with both `x₁` and `x₂`. * The rate of increase in `True ∝ FE` is higher when both `x₁` and `x₂` are closer to 1.0. * The surface is symmetrical with respect to the axes, suggesting that the function is likely symmetric in `x₁` and `x₂`. ### Interpretation The plot likely represents a mathematical function where `True ∝ FE` is proportional to the product of `x₁` and `x₂`, or some other function that exhibits similar behavior. The proportionality symbol (∝) suggests that `True ∝ FE` is not necessarily equal to a specific function of `x₁` and `x₂`, but rather varies in a similar manner. The plot could be used to visualize the relationship between two input variables (`x₁` and `x₂`) and an output variable (`True ∝ FE`). The shape of the surface provides insights into how changes in `x₁` and `x₂` affect the value of `True ∝ FE`. The smooth, continuous nature of the surface suggests a stable and predictable relationship. The use of a color gradient enhances the visualization by providing an additional dimension of information. The gradient allows for a quick and intuitive understanding of the relative values of `True ∝ FE` across the surface.

## 3D Surface Plot: Relationship Between \(x_1'\), \(x_2'\), and "True α - FE" ### Overview The image displays a three-dimensional surface plot visualizing the relationship between two independent variables, \(x_1'\) and \(x_2'\), and a dependent variable labeled "True α - FE". The plot is rendered within a 3D Cartesian coordinate system with a grid. The surface exhibits a pronounced curvature, with its color mapping indicating the magnitude of the dependent variable. ### Components/Axes * **X-axis (Bottom Right):** Labeled \(x_1'\). The scale runs from 0.0 to 1.0, with major tick marks at intervals of 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0). * **Y-axis (Bottom Left):** Labeled \(x_2'\). The scale runs from 0.0 to 1.0, with major tick marks at intervals of 0.2 (0.0, 0.2, 0.4, 0.6, 0.8, 1.0). Note: The axis is oriented such that 1.0 is on the left and 0.0 is on the right from the viewer's perspective. * **Z-axis (Vertical, Left Side):** Labeled "True α - FE". The scale runs from 0.5 to 2.5, with major tick marks at 0.5, 1.0, 1.5, 2.0, and 2.5. * **Surface & Color Mapping:** The plotted surface is colored with a gradient. The color serves as a redundant visual cue for the Z-axis value: * **Dark Purple/Blue:** Corresponds to the lowest values of "True α - FE" (approximately 0.5 to 1.0). * **Teal/Green:** Corresponds to mid-range values (approximately 1.0 to 2.0). * **Yellow:** Corresponds to the highest values (approximately 2.0 to 2.5 and above). * **Legend:** There is no separate, discrete legend box. The color gradient itself acts as a continuous legend for the Z-axis values. ### Detailed Analysis * **Spatial Grounding & Trend Verification:** The surface forms a curved sheet that rises dramatically from one corner of the \(x_1'\)-\(x_2'\) plane. * **Lowest Region:** The surface is lowest (dark purple, Z ≈ 0.5-0.8) in the region where \(x_1'\) is low (near 0.0) and \(x_2'\) is high (near 1.0). This is the front-left corner of the plotted domain. * **Highest Region:** The surface peaks (bright yellow, Z > 2.5) at the opposite corner, where \(x_1'\) is high (1.0) and \(x_2'\) is low (0.0). This is the back-right corner of the domain. * **Primary Trend:** The value of "True α - FE" increases sharply and non-linearly as \(x_1'\) increases and \(x_2'\) decreases. The gradient is steepest along the diagonal from the (0.0, 1.0) corner to the (1.0, 0.0) corner. * **Secondary Trend:** Holding \(x_2'\) constant, "True α - FE" increases with \(x_1'\). Holding \(x_1'\) constant, "True α - FE" decreases as \(x_2'\) increases. * **Data Point Approximation (Key Corners):** * At (\(x_1'\)=0.0, \(x_2'\)=1.0): "True α - FE" ≈ 0.5 - 0.8 (estimated from the lowest purple region). * At (\(x_1'\)=1.0, \(x_2'\)=0.0): "True α - FE" ≈ 2.5+ (the peak extends beyond the top Z-axis tick). * At (\(x_1'\)=0.5, \(x_2'\)=0.5): "True α - FE" ≈ 1.2 - 1.5 (estimated from the teal region at the center of the domain). ### Key Observations 1. **Strong Interaction Effect:** The relationship is not additive. The effect of increasing \(x_1'\) is much more pronounced when \(x_2'\) is low. Similarly, the effect of decreasing \(x_2'\) is amplified when \(x_1'\) is high. 2. **Monotonic Surface:** The surface appears smooth and monotonic across the displayed domain—there are no visible local minima, maxima, or saddle points within the plotted range. 3. **Color as Intensity Guide:** The color gradient effectively highlights the region of maximum value (yellow peak) and the region of minimum value (purple base), making the overall trend immediately apparent. ### Interpretation This plot demonstrates a **synergistic or multiplicative relationship** between the normalized variables \(x_1'\) and \(x_2'\) in determining the output "True α - FE". The output is maximized when \(x_1'\) is at its maximum and \(x_2'\) is at its minimum, suggesting these two conditions work together to produce the highest value. The steep, curved surface indicates that the system being modeled is highly sensitive to changes in these parameters, especially in the region where \(x_1'\) is high and \(x_2'\) is low. Small adjustments in this region could lead to large changes in the output. Conversely, the system is less sensitive (the surface is flatter) in the region where \(x_1'\) is low and \(x_2'\) is high. Without additional context, "True α - FE" could represent an error metric (where lower is better), a performance gain (where higher is better), or a physical quantity. The plot's primary message is the **non-linear, interactive dependence** of this quantity on the two input parameters. To optimize for a high "True α - FE", one should push \(x_1'\) toward 1.0 and \(x_2'\) toward 0.0. To minimize it, the opposite conditions are required.

## 3D Surface Plot: Relationship Between Variables x₁ᵣ, x₂ᵣ, and True α - FE ### Overview The image depicts a 3D surface plot visualizing the relationship between three variables: - **x₁ᵣ** (horizontal axis, left-to-right) - **x₂ᵣ** (horizontal axis, depth) - **True α - FE** (vertical axis, z-axis) The surface transitions from a flat region near the origin to a sharply curved, upward-sloping structure. A color gradient (purple to green) indicates increasing values of **True α - FE** with higher x₂ᵣ. --- ### Components/Axes 1. **Axes Labels**: - **x₁ᵣ**: Ranges from 0.0 (left) to 1.0 (right). - **x₂ᵣ**: Ranges from 0.0 (front) to 1.0 (back). - **True α - FE**: Ranges from 0.0 (bottom) to 2.5 (top). 2. **Surface**: - A 3D grid forms the base, with a smooth, continuous surface overlay. - The surface is color-coded: - **Purple**: Low values of **True α - FE** (near the origin). - **Green**: High values of **True α - FE** (upper-right region). 3. **No Legend**: - No explicit legend is present to confirm the meaning of the color gradient. --- ### Detailed Analysis 1. **Surface Behavior**: - **Flat Region**: Near the origin (x₁ᵣ ≈ 0.0, x₂ᵣ ≈ 0.0), the surface is nearly flat, with **True α - FE** ≈ 0.0–0.5. - **Curved Region**: As x₂ᵣ increases (moving backward along the plot), the surface curves upward sharply. - At x₂ᵣ ≈ 0.8–1.0, **True α - FE** reaches ~2.5 (green region). - **x₁ᵣ Dependency**: The surface remains relatively flat along the x₁ᵣ axis (left-to-right), suggesting weak or no dependency on x₁ᵣ. 2. **Color Gradient**: - The transition from purple (low values) to green (high values) aligns with increasing **True α - FE** as x₂ᵣ increases. - No explicit scale or legend is provided to quantify the color mapping. 3. **Grid Structure**: - The 3D grid uses uniform spacing for x₁ᵣ and x₂ᵣ (0.0–1.0 in 0.2 increments). - The z-axis (True α - FE) is labeled in 0.5 increments. --- ### Key Observations 1. **Dominant Trend**: - **True α - FE** increases monotonically with x₂ᵣ, with minimal dependence on x₁ᵣ. - The surface’s curvature intensifies as x₂ᵣ approaches 1.0. 2. **Anomalies**: - No outliers or discontinuities are visible. - The absence of a legend introduces uncertainty about the exact interpretation of the color gradient. 3. **Spatial Relationships**: - The flat region near the origin suggests that **True α - FE** is negligible when both x₁ᵣ and x₂ᵣ are small. - The sharp upward curve implies a nonlinear relationship between x₂ᵣ and **True α - FE**. --- ### Interpretation 1. **Mathematical Implications**: - The plot likely represents a function **f(x₁ᵣ, x₂ᵣ) = True α - FE**, where x₂ᵣ is the primary driver of the output. - The lack of x₁ᵣ dependency suggests the function may simplify to **f(x₂ᵣ)** in this visualization. 2. **Practical Significance**: - If **True α - FE** represents a physical or engineering metric (e.g., stress, efficiency), the plot highlights a critical threshold: increasing x₂ᵣ beyond ~0.6 leads to a rapid rise in the metric. - The flat region near the origin could indicate a "baseline" or "neutral" state where the metric is minimal. 3. **Uncertainties**: - Without a legend, the exact meaning of the color gradient (e.g., whether it represents magnitude, probability, or another variable) remains ambiguous. - The absence of numerical data points or error bars limits quantitative validation of the trends. --- ### Final Notes This plot emphasizes the dominance of x₂ᵣ in determining **True α - FE**, with x₁ᵣ playing a negligible role. The sharp curvature and color gradient suggest a nonlinear, possibly exponential relationship between x₂ᵣ and the output. Further analysis (e.g., adding a legend, providing numerical data) would strengthen interpretability.