An Improved Collocation Meshless Method Based on the Variable Shaped Radial Basis Function for the Solution of the Interior Acoustic Problems

<table>The interior acoustic problem; <svg height="11.425" id="M16" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 12.9875 11.425" width="12.9875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.363)"><path d="M719 146l-18 -146h-243l-5 18q2 4 17 25t24 36l25 42t26.5 52.5t21 56.5t16 65t5.5 68q0 121 -55.5 194.5t-149.5 73.5q-95 0 -150 -72t-55 -199q0 -54 18.5 -114t36.5 -94t48.5 -82.5t32.5 -51.5l-7 -18h-244l-17 146l27 1q17 -49 31.5 -62t45.5 -13h89l-21 21t-26 26.5
l-26 29t-28 35.5l-23 37.5t-20.5 43t-12 45.5t-5.5 51q0 133 88 218.5t219 85.5q132 0 218 -83t86 -218q0 -40 -11.5 -78t-33 -71.5t-39 -56t-44 -51t-32.5 -35.5h84q35 0 48 12t31 63z" id="x3A9"></path></g>
</svg>: computational domain; <svg height="15.25" id="M17" style="vertical-align:-3.27605pt" version="1.1" viewbox="0 0 18.924999 15.25" width="18.924999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M485 501l-30 -4q-15 64 -35 90q-12 16 -33.5 22t-68.5 6h-74q-21 0 -26.5 -5t-5.5 -26v-458q0 -62 15.5 -78t83.5 -20v-28h-270v28q58 4 72 20t14 78v399q0 61 -13.5 77t-71.5 20v28h433q0 -51 10 -149z" id="x393"></path></g>
<g transform="matrix(.012,0,0,-.012,8.675,15.188)"><path d="M780 372q0 -191 -178 -297q-128 -75 -335 -75h-250l7 27q61 7 77 22t28 76l77 398q11 62 0 76.5t-77 22.5l6 28h292q164 0 258.5 -67.5t94.5 -210.5zM678 375q0 119 -73 180.5t-201 61.5q-62 0 -82 -13q-17 -7 -25 -53l-80 -427q-8 -47 -4 -62t26 -20q24 -6 80 -6
q166 0 262.5 95t96.5 244z" id="x1D437"></path></g>
</svg>: sound soft boundary; <svg height="15.35" id="M18" style="vertical-align:-3.3525pt" version="1.1" viewbox="0 0 19.762501 15.35" width="19.762501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M485 501l-30 -4q-15 64 -35 90q-12 16 -33.5 22t-68.5 6h-74q-21 0 -26.5 -5t-5.5 -26v-458q0 -62 15.5 -78t83.5 -20v-28h-270v28q58 4 72 20t14 78v399q0 61 -13.5 77t-71.5 20v28h433q0 -51 10 -149z" id="x393"></path></g>
<g transform="matrix(.012,0,0,-.012,8.675,15.188)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
</svg>: sound hard boundary; <svg height="15.35" id="M19" style="vertical-align:-3.3525pt" version="1.1" viewbox="0 0 17.025 15.35" width="17.025" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.113)"><path d="M485 501l-30 -4q-15 64 -35 90q-12 16 -33.5 22t-68.5 6h-74q-21 0 -26.5 -5t-5.5 -26v-458q0 -62 15.5 -78t83.5 -20v-28h-270v28q58 4 72 20t14 78v399q0 61 -13.5 77t-71.5 20v28h433q0 -51 10 -149z" id="x393"></path></g>
<g transform="matrix(.012,0,0,-.012,8.675,15.188)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg>: impedance boundary. </table>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: An Improved Collocation Meshless Method Based on the Variable Shaped Radial Basis Function for the Solution of the Interior Acoustic Problems 

Figure 1 | An Improved Collocation Meshless Method Based on the Variable Shaped Radial Basis Function for the Solution of the Interior Acoustic Problems 