Mathematical Problems in Engineering

Research Article

Shape-Free Finite Element Method: Another Way between Mesh and Mesh-Free Methods

Table 4

Results at selected locations for the linear bending problem (Figure 5).


		Q8 (3 × 3)	HSF-Q8-15 (5 × 5)	US-QUAD8 (3 × 3)	US-ATFQ8 (5 × 5 or 3 × 3)	Exact

	()	−66.871	−120.00	−66.871	−120.000	−120.0
Mesh 1	()	66.871	120.00	66.871	120.000	120.0
	v() × 10³	6.282	8.558	6.282	8.0460	8.046

	()	−35.248	−113.63	−74.317	−120.000	−120.0
Mesh 2	()	67.040	119.63	71.781	120.000	120.0
	v() × 10³	1.611	7.104	6.415	8.0460	8.046

	()	−11.307	−68.645	−66.349	−120.000	−120.0
	()	−55.699	−66.980	−67.974	−120.000	−120.0
Mesh 3	()	55.305	66.146	67.995	120.000	120.0
	()	11.073	69.328	66.328	120.000	120.0
	v() × 10³	4.520	6.350	6.292	8.0460	8.046

	()		−72.471		−120.00	−120.0
Mesh 4	()	Failed	76.265	Failed	120.00	120.0
	v() × 10³		6.663		8.0460	8.046

	()		−77.515	−58.158	−120.000	−120.0
Mesh 5	()	Failed	66.306	59.158	120.000	120.0
	v() × 10³		6.729	5.471	8.0460	8.046

	()		−70.346	−42.671	−120.000	−120.0
	()		−66.835	−44.636	−120.000	−120.0
Mesh 6	()	Failed	66.877	40.974	120.000	120.0
	()		68.512	41.048	120.000	120.0
	v( × 10³		6.373	3.752	8.0460	8.046

	()	−105.69	−120.00	−105.69	−120.000	−120.0
Mesh 7	()	105.69	120.00	105.69	120.000	120.0
	v() × 10⁴	3.485	3.975	3.485	3.6600	3.66

	()	−115.25	−119.02	−118.99	−120.000	−120.0
Mesh 8	()	115.25	119.02	118.99	120.000	120.0
	v() × 10⁴	3.3423	3.796	3.5570	3.6600	3.66

	()	−118.68	−120.00	−118.68	−120.000	−120.0
Mesh 9	()	118.68	120.00	118.68	120.000	120.0
	v() × 10⁵	11.1963	15.750	11.1963	13.200	13.2

	()	−97.066	−121.61	−118.94	−120.00 ^b/−119.65^a	−120.0
Mesh 10	()	138.21	124.61	117.74	120.00 ^b/119.36^a	120.0
	v() × 10⁵	12.8162	13.738	12.9157	13.200 ^b/13.190^a	13.2

	()		−24.732	−36.104	−120.000	−120.0
Mesh 11	()	Failed	20.048	36.230	120.000	120.0
	v() × 10³		1.517	3.1470	8.0460	8.046

3 × 3 integration scheme; ^b5 × 5 integration scheme.