That pattern seems strange for Dog Urine. Have you checked for Cutworm or White Grubs. Fence in a 6'x6' or similar area and cover with ground cover cloth then 1" crushed stone. Train (make) the dogs do their business there. Get a few rolls of sod. Soak each in a 5 gal pail of water for a few minutes and cut out patches. Dig out dead grass, place on new sod, cut matching patch and plant.