Meshing, Numerical Integration and Convergence
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Meshing, Numerical Integration and Convergence.
§4.1 Geometry idealization and local detail
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `error=Q_FE-Q_reference`. **Field study:** Compare a smooth plate and a notched plate under identical remote traction.
§4.2 Element quality and Jacobians
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `det(J)>0`. **Field study:** Compute a triangular area and flag zero-area elements.
§4.3 Gauss quadrature and numerical integration
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `K_e ≈ ∑ w_g B_gᵀ D B_g det(J_g)`. **Field study:** Check an element stiffness integral under one- versus two-point quadrature.
§4.4 h-refinement, p-refinement and adaptivity
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `Q_h=Q_*+C h^p`. **Field study:** Generate three mesh levels and examine a displacement convergence curve.
§4.5 Stress singularities and extrapolation
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `σ(r)∼ r^{λ-1}`. **Field study:** Study a sharp notch, then add a physical fillet and compare convergence.
Step-by-Step Finite Element Analysis and Computational Mechanics Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **uniform element size**, based on `h=L/n`. Illustrative inputs are **Domain length=1 m, Number of elements=20 —**. Converting the displayed units into SI inside the calculation gives **50 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **uniform element size**, based on `h=L/n`. Illustrative inputs are **Domain length=1 m, Number of elements=20 —**. Converting the displayed units into SI inside the calculation gives **50 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **uniform element size**, based on `h=L/n`. Illustrative inputs are **Domain length=1 m, Number of elements=20 —**. Converting the displayed units into SI inside the calculation gives **50 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **richardson extrapolation**, based on `Q*=Q_f+(Q_f-Q_c)/(r^p-1)`. Illustrative inputs are **Fine-grid displacement=1.96 mm, Coarse-grid displacement=1.84 mm, Refinement ratio=2 —, Assumed convergence order=2 —**. Converting the displayed units into SI inside the calculation gives **2 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **richardson extrapolation**, based on `Q*=Q_f+(Q_f-Q_c)/(r^p-1)`. Illustrative inputs are **Fine-grid displacement=1.96 mm, Coarse-grid displacement=1.84 mm, Refinement ratio=2 —, Assumed convergence order=2 —**. Converting the displayed units into SI inside the calculation gives **2 mm**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.